poimirlem4

Description: Lemma for poimir connecting the admissible faces on the back face of the ( M + 1 ) -cube to admissible simplices in the M -cube. (Contributed by Brendan Leahy, 21-Aug-2020)

	Ref	Expression
Hypotheses	poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	poimirlem4.1	⊢ ( 𝜑 → 𝐾 ∈ ℕ )
	poimirlem4.2	⊢ ( 𝜑 → 𝑀 ∈ ℕ₀ )
	poimirlem4.3	⊢ ( 𝜑 → 𝑀 < 𝑁 )
Assertion	poimirlem4	⊢ ( 𝜑 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ≈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } )

Proof

Step	Hyp	Ref	Expression
1		poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		poimirlem4.1	⊢ ( 𝜑 → 𝐾 ∈ ℕ )
3		poimirlem4.2	⊢ ( 𝜑 → 𝑀 ∈ ℕ₀ )
4		poimirlem4.3	⊢ ( 𝜑 → 𝑀 < 𝑁 )
5	1	adantr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → 𝑁 ∈ ℕ )
6	2	adantr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → 𝐾 ∈ ℕ )
7	3	adantr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → 𝑀 ∈ ℕ₀ )
8	4	adantr	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → 𝑀 < 𝑁 )
9		xp1st	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 1^st ‘ 𝑡 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) )
10		elmapi	⊢ ( ( 1^st ‘ 𝑡 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) → ( 1^st ‘ 𝑡 ) : ( 1 ... 𝑀 ) ⟶ ( 0 ..^ 𝐾 ) )
11	9 10	syl	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 1^st ‘ 𝑡 ) : ( 1 ... 𝑀 ) ⟶ ( 0 ..^ 𝐾 ) )
12	11	adantl	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( 1^st ‘ 𝑡 ) : ( 1 ... 𝑀 ) ⟶ ( 0 ..^ 𝐾 ) )
13		xp2nd	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 2^nd ‘ 𝑡 ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } )
14		fvex	⊢ ( 2^nd ‘ 𝑡 ) ∈ V
15		f1oeq1	⊢ ( 𝑓 = ( 2^nd ‘ 𝑡 ) → ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ↔ ( 2^nd ‘ 𝑡 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ) )
16	14 15	elab	⊢ ( ( 2^nd ‘ 𝑡 ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ↔ ( 2^nd ‘ 𝑡 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
17	13 16	sylib	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 2^nd ‘ 𝑡 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
18	17	adantl	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( 2^nd ‘ 𝑡 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
19	5 6 7 8 12 18	poimirlem3	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 → ( ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ) )
20		fvex	⊢ ( 1^st ‘ 𝑡 ) ∈ V
21		snex	⊢ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ∈ V
22	20 21	unex	⊢ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∈ V
23		snex	⊢ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ∈ V
24	14 23	unex	⊢ ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ∈ V
25	22 24	op1std	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( 1^st ‘ 𝑠 ) = ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
26	22 24	op2ndd	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( 2^nd ‘ 𝑠 ) = ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
27	26	imaeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) = ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) )
28	27	xpeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
29	26	imaeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) = ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
30	29	xpeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) )
31	28 30	uneq12d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) )
32	25 31	oveq12d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) = ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) )
33	32	uneq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
34	33	csbeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
35	34	eqeq2d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
36	35	rexbidv	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
37	36	ralbidv	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
38	25	fveq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ‘ ( 𝑀 + 1 ) ) )
39	38	eqeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ↔ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ‘ ( 𝑀 + 1 ) ) = 0 ) )
40	26	fveq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ‘ ( 𝑀 + 1 ) ) )
41	40	eqeq1d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ↔ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) )
42	37 39 41	3anbi123d	⊢ ( 𝑠 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ↔ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) )
43	42	elrab	⊢ ( ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ↔ ( ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) )
44	19 43	imbitrrdi	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 → ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ) )
45	44	ralrimiva	⊢ ( 𝜑 → ∀ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 → ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ) )
46		fveq2	⊢ ( 𝑠 = 𝑡 → ( 1^st ‘ 𝑠 ) = ( 1^st ‘ 𝑡 ) )
47		fveq2	⊢ ( 𝑠 = 𝑡 → ( 2^nd ‘ 𝑠 ) = ( 2^nd ‘ 𝑡 ) )
48	47	imaeq1d	⊢ ( 𝑠 = 𝑡 → ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) )
49	48	xpeq1d	⊢ ( 𝑠 = 𝑡 → ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
50	47	imaeq1d	⊢ ( 𝑠 = 𝑡 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) = ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
51	50	xpeq1d	⊢ ( 𝑠 = 𝑡 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) )
52	49 51	uneq12d	⊢ ( 𝑠 = 𝑡 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) )
53	46 52	oveq12d	⊢ ( 𝑠 = 𝑡 → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) )
54	53	uneq1d	⊢ ( 𝑠 = 𝑡 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) )
55	54	csbeq1d	⊢ ( 𝑠 = 𝑡 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
56	55	eqeq2d	⊢ ( 𝑠 = 𝑡 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
57	56	rexbidv	⊢ ( 𝑠 = 𝑡 → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
58	57	ralbidv	⊢ ( 𝑠 = 𝑡 → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
59	58	ralrab	⊢ ( ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ↔ ∀ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 → ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ) )
60	45 59	sylibr	⊢ ( 𝜑 → ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } )
61		xp1st	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 1^st ‘ 𝑘 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) )
62		elmapi	⊢ ( ( 1^st ‘ 𝑘 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) → ( 1^st ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) ⟶ ( 0 ..^ 𝐾 ) )
63	61 62	syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 1^st ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) ⟶ ( 0 ..^ 𝐾 ) )
64		fzssp1	⊢ ( 1 ... 𝑀 ) ⊆ ( 1 ... ( 𝑀 + 1 ) )
65		fssres	⊢ ( ( ( 1^st ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) ⟶ ( 0 ..^ 𝐾 ) ∧ ( 1 ... 𝑀 ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) ⟶ ( 0 ..^ 𝐾 ) )
66	63 64 65	sylancl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) ⟶ ( 0 ..^ 𝐾 ) )
67		ovex	⊢ ( 0 ..^ 𝐾 ) ∈ V
68		ovex	⊢ ( 1 ... 𝑀 ) ∈ V
69	67 68	elmap	⊢ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) ↔ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) ⟶ ( 0 ..^ 𝐾 ) )
70	66 69	sylibr	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) )
71	70	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) )
72		xp2nd	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 2^nd ‘ 𝑘 ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } )
73		fvex	⊢ ( 2^nd ‘ 𝑘 ) ∈ V
74		f1oeq1	⊢ ( 𝑓 = ( 2^nd ‘ 𝑘 ) → ( 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) ↔ ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) ) )
75	73 74	elab	⊢ ( ( 2^nd ‘ 𝑘 ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ↔ ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) )
76	72 75	sylib	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) )
77		f1of1	⊢ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) → ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1→ ( 1 ... ( 𝑀 + 1 ) ) )
78	76 77	syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1→ ( 1 ... ( 𝑀 + 1 ) ) )
79		f1ores	⊢ ( ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1→ ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 1 ... 𝑀 ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) ) → ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) )
80	78 64 79	sylancl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) )
81	80	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) )
82		dff1o3	⊢ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) ↔ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –onto→ ( 1 ... ( 𝑀 + 1 ) ) ∧ Fun ^◡ ( 2^nd ‘ 𝑘 ) ) )
83	82	simprbi	⊢ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) → Fun ^◡ ( 2^nd ‘ 𝑘 ) )
84		imadif	⊢ ( Fun ^◡ ( 2^nd ‘ 𝑘 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) )
85	76 83 84	3syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) )
86	85	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) )
87		f1ofo	⊢ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) → ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –onto→ ( 1 ... ( 𝑀 + 1 ) ) )
88		foima	⊢ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –onto→ ( 1 ... ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) = ( 1 ... ( 𝑀 + 1 ) ) )
89	76 87 88	3syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) = ( 1 ... ( 𝑀 + 1 ) ) )
90	89	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) = ( 1 ... ( 𝑀 + 1 ) ) )
91		f1ofn	⊢ ( ( 2^nd ‘ 𝑘 ) : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) → ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
92	76 91	syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
93		nn0p1nn	⊢ ( 𝑀 ∈ ℕ₀ → ( 𝑀 + 1 ) ∈ ℕ )
94	3 93	syl	⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ℕ )
95		elfz1end	⊢ ( ( 𝑀 + 1 ) ∈ ℕ ↔ ( 𝑀 + 1 ) ∈ ( 1 ... ( 𝑀 + 1 ) ) )
96	94 95	sylib	⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ( 1 ... ( 𝑀 + 1 ) ) )
97		fnsnfv	⊢ ( ( ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( 1 ... ( 𝑀 + 1 ) ) ) → { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } = ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) )
98	92 96 97	syl2anr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } = ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) )
99		sneq	⊢ ( ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) → { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } = { ( 𝑀 + 1 ) } )
100	98 99	sylan9req	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) = { ( 𝑀 + 1 ) } )
101	90 100	difeq12d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) = ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) )
102	86 101	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) )
103		1z	⊢ 1 ∈ ℤ
104		nn0uz	⊢ ℕ₀ = ( ℤ_≥ ‘ 0 )
105		1m1e0	⊢ ( 1 − 1 ) = 0
106	105	fveq2i	⊢ ( ℤ_≥ ‘ ( 1 − 1 ) ) = ( ℤ_≥ ‘ 0 )
107	104 106	eqtr4i	⊢ ℕ₀ = ( ℤ_≥ ‘ ( 1 − 1 ) )
108	3 107	eleqtrdi	⊢ ( 𝜑 → 𝑀 ∈ ( ℤ_≥ ‘ ( 1 − 1 ) ) )
109		fzsuc2	⊢ ( ( 1 ∈ ℤ ∧ 𝑀 ∈ ( ℤ_≥ ‘ ( 1 − 1 ) ) ) → ( 1 ... ( 𝑀 + 1 ) ) = ( ( 1 ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) )
110	103 108 109	sylancr	⊢ ( 𝜑 → ( 1 ... ( 𝑀 + 1 ) ) = ( ( 1 ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) )
111	110	difeq1d	⊢ ( 𝜑 → ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) = ( ( ( 1 ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) ∖ { ( 𝑀 + 1 ) } ) )
112		difun2	⊢ ( ( ( 1 ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) ∖ { ( 𝑀 + 1 ) } ) = ( ( 1 ... 𝑀 ) ∖ { ( 𝑀 + 1 ) } )
113	111 112	eqtrdi	⊢ ( 𝜑 → ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) = ( ( 1 ... 𝑀 ) ∖ { ( 𝑀 + 1 ) } ) )
114	3	nn0red	⊢ ( 𝜑 → 𝑀 ∈ ℝ )
115		ltp1	⊢ ( 𝑀 ∈ ℝ → 𝑀 < ( 𝑀 + 1 ) )
116		peano2re	⊢ ( 𝑀 ∈ ℝ → ( 𝑀 + 1 ) ∈ ℝ )
117		ltnle	⊢ ( ( 𝑀 ∈ ℝ ∧ ( 𝑀 + 1 ) ∈ ℝ ) → ( 𝑀 < ( 𝑀 + 1 ) ↔ ¬ ( 𝑀 + 1 ) ≤ 𝑀 ) )
118	116 117	mpdan	⊢ ( 𝑀 ∈ ℝ → ( 𝑀 < ( 𝑀 + 1 ) ↔ ¬ ( 𝑀 + 1 ) ≤ 𝑀 ) )
119	115 118	mpbid	⊢ ( 𝑀 ∈ ℝ → ¬ ( 𝑀 + 1 ) ≤ 𝑀 )
120	114 119	syl	⊢ ( 𝜑 → ¬ ( 𝑀 + 1 ) ≤ 𝑀 )
121		elfzle2	⊢ ( ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) → ( 𝑀 + 1 ) ≤ 𝑀 )
122	120 121	nsyl	⊢ ( 𝜑 → ¬ ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) )
123		difsn	⊢ ( ¬ ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) → ( ( 1 ... 𝑀 ) ∖ { ( 𝑀 + 1 ) } ) = ( 1 ... 𝑀 ) )
124	122 123	syl	⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∖ { ( 𝑀 + 1 ) } ) = ( 1 ... 𝑀 ) )
125	113 124	eqtrd	⊢ ( 𝜑 → ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) = ( 1 ... 𝑀 ) )
126	125	imaeq2d	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) )
127	126	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) )
128	125	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) = ( 1 ... 𝑀 ) )
129	102 127 128	3eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) = ( 1 ... 𝑀 ) )
130	129	f1oeq3d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑀 ) ) ↔ ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ) )
131	81 130	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
132	73	resex	⊢ ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ V
133		f1oeq1	⊢ ( 𝑓 = ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) → ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ↔ ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ) )
134	132 133	elab	⊢ ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ↔ ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
135	131 134	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } )
136	71 135	opelxpd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) )
137	136	3ad2antr3	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) )
138		3anass	⊢ ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ↔ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) )
139	138	biancomi	⊢ ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ↔ ( ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
140	94	nnzd	⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ℤ )
141		uzid	⊢ ( ( 𝑀 + 1 ) ∈ ℤ → ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) )
142		peano2uz	⊢ ( ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) → ( ( 𝑀 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) )
143	140 141 142	3syl	⊢ ( 𝜑 → ( ( 𝑀 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) )
144	3	nn0zd	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
145	1	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
146		zltp1le	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )
147		peano2z	⊢ ( 𝑀 ∈ ℤ → ( 𝑀 + 1 ) ∈ ℤ )
148		eluz	⊢ ( ( ( 𝑀 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )
149	147 148	sylan	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )
150	146 149	bitr4d	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) ) )
151	144 145 150	syl2anc	⊢ ( 𝜑 → ( 𝑀 < 𝑁 ↔ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) ) )
152	4 151	mpbid	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) )
153		fzsplit2	⊢ ( ( ( ( 𝑀 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝑀 + 1 ) ... 𝑁 ) = ( ( ( 𝑀 + 1 ) ... ( 𝑀 + 1 ) ) ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) )
154	143 152 153	syl2anc	⊢ ( 𝜑 → ( ( 𝑀 + 1 ) ... 𝑁 ) = ( ( ( 𝑀 + 1 ) ... ( 𝑀 + 1 ) ) ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) )
155		fzsn	⊢ ( ( 𝑀 + 1 ) ∈ ℤ → ( ( 𝑀 + 1 ) ... ( 𝑀 + 1 ) ) = { ( 𝑀 + 1 ) } )
156	140 155	syl	⊢ ( 𝜑 → ( ( 𝑀 + 1 ) ... ( 𝑀 + 1 ) ) = { ( 𝑀 + 1 ) } )
157	156	uneq1d	⊢ ( 𝜑 → ( ( ( 𝑀 + 1 ) ... ( 𝑀 + 1 ) ) ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) = ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) )
158	154 157	eqtrd	⊢ ( 𝜑 → ( ( 𝑀 + 1 ) ... 𝑁 ) = ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) )
159	158	xpeq1d	⊢ ( 𝜑 → ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) = ( ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
160	159	uneq2d	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
161		xpundir	⊢ ( ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( { ( 𝑀 + 1 ) } × { 0 } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) )
162		ovex	⊢ ( 𝑀 + 1 ) ∈ V
163		c0ex	⊢ 0 ∈ V
164	162 163	xpsn	⊢ ( { ( 𝑀 + 1 ) } × { 0 } ) = { ⟨ ( 𝑀 + 1 ) , 0 ⟩ }
165	164	uneq1i	⊢ ( ( { ( 𝑀 + 1 ) } × { 0 } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) )
166	161 165	eqtri	⊢ ( ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) = ( { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) )
167	166	uneq2i	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
168		unass	⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
169	167 168	eqtr4i	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( { ( 𝑀 + 1 ) } ∪ ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) )
170	160 169	eqtrdi	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
171	170	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
172	162	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑀 + 1 ) ∈ V )
173	163	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 0 ∈ V )
174	110	eqcomd	⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) = ( 1 ... ( 𝑀 + 1 ) ) )
175	174	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1 ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) = ( 1 ... ( 𝑀 + 1 ) ) )
176		fveq2	⊢ ( 𝑛 = ( 𝑀 + 1 ) → ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) = ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) )
177		fveq2	⊢ ( 𝑛 = ( 𝑀 + 1 ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) )
178	176 177	oveq12d	⊢ ( 𝑛 = ( 𝑀 + 1 ) → ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) ) )
179		simplrl	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 )
180		imain	⊢ ( Fun ^◡ ( 2^nd ‘ 𝑘 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) )
181	76 83 180	3syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) )
182	181	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) )
183		elfznn0	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℕ₀ )
184	183	nn0red	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℝ )
185	184	ltp1d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 < ( 𝑗 + 1 ) )
186		fzdisj	⊢ ( 𝑗 < ( 𝑗 + 1 ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) = ∅ )
187	185 186	syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) = ∅ )
188	187	imaeq2d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( ( 2^nd ‘ 𝑘 ) “ ∅ ) )
189		ima0	⊢ ( ( 2^nd ‘ 𝑘 ) “ ∅ ) = ∅
190	188 189	eqtrdi	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ∅ )
191	182 190	sylan9req	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ∅ )
192		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) )
193	92	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
194		nn0p1nn	⊢ ( 𝑗 ∈ ℕ₀ → ( 𝑗 + 1 ) ∈ ℕ )
195		nnuz	⊢ ℕ = ( ℤ_≥ ‘ 1 )
196	194 195	eleqtrdi	⊢ ( 𝑗 ∈ ℕ₀ → ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
197		fzss1	⊢ ( ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) )
198	183 196 197	3syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) )
199		elfzuz3	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ( ℤ_≥ ‘ 𝑗 ) )
200		eluzp1p1	⊢ ( 𝑀 ∈ ( ℤ_≥ ‘ 𝑗 ) → ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑗 + 1 ) ) )
201		eluzfz2	⊢ ( ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑗 + 1 ) ) → ( 𝑀 + 1 ) ∈ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) )
202	199 200 201	3syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑀 + 1 ) ∈ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) )
203	198 202	jca	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
204		fnfvima	⊢ ( ( ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) ∧ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) → ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
205	204	3expb	⊢ ( ( ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) ∧ ( ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) → ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
206	193 203 205	syl2an	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
207	192 206	eqeltrrd	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑀 + 1 ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
208		1ex	⊢ 1 ∈ V
209		fnconstg	⊢ ( 1 ∈ V → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) )
210	208 209	ax-mp	⊢ ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) )
211		fnconstg	⊢ ( 0 ∈ V → ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
212	163 211	ax-mp	⊢ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) )
213		fvun2	⊢ ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ∧ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ∅ ∧ ( 𝑀 + 1 ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ‘ ( 𝑀 + 1 ) ) )
214	210 212 213	mp3an12	⊢ ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ∅ ∧ ( 𝑀 + 1 ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ‘ ( 𝑀 + 1 ) ) )
215	191 207 214	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ‘ ( 𝑀 + 1 ) ) )
216	163	fvconst2	⊢ ( ( 𝑀 + 1 ) ∈ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ‘ ( 𝑀 + 1 ) ) = 0 )
217	207 216	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ‘ ( 𝑀 + 1 ) ) = 0 )
218	215 217	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) = 0 )
219	218	adantlrl	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) = 0 )
220	179 219	oveq12d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) ) = ( 0 + 0 ) )
221		00id	⊢ ( 0 + 0 ) = 0
222	220 221	eqtrdi	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ ( 𝑀 + 1 ) ) ) = 0 )
223	178 222	sylan9eqr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 = ( 𝑀 + 1 ) ) → ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) = 0 )
224	172 173 175 223	fmptapd	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
225	224	uneq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
226	171 225	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
227		elmapfn	⊢ ( ( 1^st ‘ 𝑘 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) → ( 1^st ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
228	61 227	syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( 1^st ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
229		fnssres	⊢ ( ( ( 1^st ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 1 ... 𝑀 ) ⊆ ( 1 ... ( 𝑀 + 1 ) ) ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) Fn ( 1 ... 𝑀 ) )
230	228 64 229	sylancl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) Fn ( 1 ... 𝑀 ) )
231	230	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) Fn ( 1 ... 𝑀 ) )
232		fnconstg	⊢ ( 0 ∈ V → ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
233	163 232	ax-mp	⊢ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) )
234	210 233	pm3.2i	⊢ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
235		imain	⊢ ( Fun ^◡ ( 2^nd ‘ 𝑘 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
236	76 83 235	3syl	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
237		fzdisj	⊢ ( 𝑗 < ( 𝑗 + 1 ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑀 ) ) = ∅ )
238	185 237	syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑀 ) ) = ∅ )
239	238	imaeq2d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ( ( 2^nd ‘ 𝑘 ) “ ∅ ) )
240	239 189	eqtrdi	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ∅ )
241	236 240	sylan9req	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ∅ )
242		fnun	⊢ ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ∅ ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
243	234 241 242	sylancr	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
244	243	ad4ant24	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
245	101	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) = ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) )
246	85	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) )
247	183 194	syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 + 1 ) ∈ ℕ )
248	247 195	eleqtrdi	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
249		fzsplit2	⊢ ( ( ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑀 ∈ ( ℤ_≥ ‘ 𝑗 ) ) → ( 1 ... 𝑀 ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
250	248 199 249	syl2anc	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 1 ... 𝑀 ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
251	128 250	sylan9eq	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
252	251	imaeq2d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
253	246 252	eqtr3d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) ∖ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
254	125	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) = ( 1 ... 𝑀 ) )
255	245 253 254	3eqtr3rd	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 1 ... 𝑀 ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
256		imaundi	⊢ ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
257	255 256	eqtrdi	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 1 ... 𝑀 ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) )
258	257	fneq2d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ↔ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ) ) )
259	244 258	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) )
260		fzss2	⊢ ( 𝑀 ∈ ( ℤ_≥ ‘ 𝑗 ) → ( 1 ... 𝑗 ) ⊆ ( 1 ... 𝑀 ) )
261		resima2	⊢ ( ( 1 ... 𝑗 ) ⊆ ( 1 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) )
262	199 260 261	3syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) )
263	262	xpeq1d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
264	183 196	syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
265		fzss1	⊢ ( ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑗 + 1 ) ... 𝑀 ) ⊆ ( 1 ... 𝑀 ) )
266		resima2	⊢ ( ( ( 𝑗 + 1 ) ... 𝑀 ) ⊆ ( 1 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
267	264 265 266	3syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
268	267	xpeq1d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) )
269	263 268	uneq12d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) )
270	269	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) )
271	270	fneq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ↔ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ) )
272	259 271	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) )
273		fzfid	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 1 ... 𝑀 ) ∈ Fin )
274		inidm	⊢ ( ( 1 ... 𝑀 ) ∩ ( 1 ... 𝑀 ) ) = ( 1 ... 𝑀 )
275		fvres	⊢ ( 𝑛 ∈ ( 1 ... 𝑀 ) → ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ‘ 𝑛 ) = ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) )
276	275	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ‘ 𝑛 ) = ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) )
277		disjsn	⊢ ( ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ ↔ ¬ ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) )
278	122 277	sylibr	⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ )
279	278	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ )
280	259 279	jca	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ∧ ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ ) )
281		fnconstg	⊢ ( 0 ∈ V → ( { ( 𝑀 + 1 ) } × { 0 } ) Fn { ( 𝑀 + 1 ) } )
282	163 281	ax-mp	⊢ ( { ( 𝑀 + 1 ) } × { 0 } ) Fn { ( 𝑀 + 1 ) }
283		fvun1	⊢ ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ∧ ( { ( 𝑀 + 1 ) } × { 0 } ) Fn { ( 𝑀 + 1 ) } ∧ ( ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) )
284	282 283	mp3an2	⊢ ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ∧ ( ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) )
285	284	anassrs	⊢ ( ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) Fn ( 1 ... 𝑀 ) ∧ ( ( 1 ... 𝑀 ) ∩ { ( 𝑀 + 1 ) } ) = ∅ ) ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) )
286	280 285	sylan	⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) )
287	247	nnzd	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 + 1 ) ∈ ℤ )
288	183	nn0cnd	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℂ )
289		pncan1	⊢ ( 𝑗 ∈ ℂ → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 )
290	288 289	syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 )
291	290	fveq2d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ℤ_≥ ‘ ( ( 𝑗 + 1 ) − 1 ) ) = ( ℤ_≥ ‘ 𝑗 ) )
292	199 291	eleqtrrd	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ( ℤ_≥ ‘ ( ( 𝑗 + 1 ) − 1 ) ) )
293		fzsuc2	⊢ ( ( ( 𝑗 + 1 ) ∈ ℤ ∧ 𝑀 ∈ ( ℤ_≥ ‘ ( ( 𝑗 + 1 ) − 1 ) ) ) → ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) = ( ( ( 𝑗 + 1 ) ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) )
294	287 292 293	syl2anc	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) = ( ( ( 𝑗 + 1 ) ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) )
295	294	imaeq2d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( ( 𝑗 + 1 ) ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) ) )
296		imaundi	⊢ ( ( 2^nd ‘ 𝑘 ) “ ( ( ( 𝑗 + 1 ) ... 𝑀 ) ∪ { ( 𝑀 + 1 ) } ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) )
297	295 296	eqtrdi	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) )
298	297	xpeq1d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) × { 0 } ) )
299		xpundir	⊢ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) )
300	298 299	eqtrdi	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) )
301	300	uneq2d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) ) )
302		unass	⊢ ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) )
303	301 302	eqtr4di	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) )
304	303	adantl	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) )
305	98	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } × { 0 } ) = ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) )
306	305	uneq2d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) )
307	306	adantr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ { ( 𝑀 + 1 ) } ) × { 0 } ) ) )
308	304 307	eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } × { 0 } ) ) )
309	99	xpeq1d	⊢ ( ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) → ( { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } × { 0 } ) = ( { ( 𝑀 + 1 ) } × { 0 } ) )
310	309	uneq2d	⊢ ( ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) } × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) )
311	308 310	sylan9eq	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) )
312	311	an32s	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) )
313	312	fveq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
314	313	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ∪ ( { ( 𝑀 + 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
315	269	fveq1d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) )
316	315	ad2antlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) )
317	286 314 316	3eqtr4rd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑀 ) ) → ( ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) )
318	231 272 273 273 274 276 317	offval	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
319	318	uneq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) )
320	319	adantlrl	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... 𝑀 ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) )
321	228	adantr	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 1^st ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
322	210 212	pm3.2i	⊢ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
323	181 190	sylan9req	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ∅ )
324		fnun	⊢ ( ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) ∧ ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∩ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ∅ ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) )
325	322 323 324	sylancr	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) )
326		peano2uz	⊢ ( 𝑀 ∈ ( ℤ_≥ ‘ 𝑗 ) → ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ 𝑗 ) )
327	199 326	syl	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ 𝑗 ) )
328		fzsplit2	⊢ ( ( ( 𝑗 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ ( 𝑀 + 1 ) ∈ ( ℤ_≥ ‘ 𝑗 ) ) → ( 1 ... ( 𝑀 + 1 ) ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
329	264 327 328	syl2anc	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 1 ... ( 𝑀 + 1 ) ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
330	329	imaeq2d	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) )
331		imaundi	⊢ ( ( 2^nd ‘ 𝑘 ) “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
332	330 331	eqtr2di	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... ( 𝑀 + 1 ) ) ) )
333	332 89	sylan9eqr	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) = ( 1 ... ( 𝑀 + 1 ) ) )
334	333	fneq2d	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) ∪ ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) ) ↔ ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) Fn ( 1 ... ( 𝑀 + 1 ) ) ) )
335	325 334	mpbid	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) Fn ( 1 ... ( 𝑀 + 1 ) ) )
336		fzfid	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 1 ... ( 𝑀 + 1 ) ) ∈ Fin )
337		inidm	⊢ ( ( 1 ... ( 𝑀 + 1 ) ) ∩ ( 1 ... ( 𝑀 + 1 ) ) ) = ( 1 ... ( 𝑀 + 1 ) )
338		eqidd	⊢ ( ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ) → ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) = ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) )
339		eqidd	⊢ ( ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ) → ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) )
340	321 335 336 336 337 338 339	offval	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
341	340	uneq1d	⊢ ( ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
342	341	ad4ant24	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 𝑛 ∈ ( 1 ... ( 𝑀 + 1 ) ) ↦ ( ( ( 1^st ‘ 𝑘 ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
343	226 320 342	3eqtr4rd	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) )
344	343	csbeq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
345	344	eqeq2d	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
346	345	rexbidva	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
347	346	ralbidv	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
348	347	biimpd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 → ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
349	348	impr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) ) → ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
350	139 349	sylan2b	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
351		1st2nd2	⊢ ( 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) → 𝑘 = ⟨ ( 1^st ‘ 𝑘 ) , ( 2^nd ‘ 𝑘 ) ⟩ )
352	351	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → 𝑘 = ⟨ ( 1^st ‘ 𝑘 ) , ( 2^nd ‘ 𝑘 ) ⟩ )
353		fnsnsplit	⊢ ( ( ( 1^st ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( 1 ... ( 𝑀 + 1 ) ) ) → ( 1^st ‘ 𝑘 ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) )
354	228 96 353	syl2anr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( 1^st ‘ 𝑘 ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) )
355	354	adantr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ) → ( 1^st ‘ 𝑘 ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) )
356	125	reseq2d	⊢ ( 𝜑 → ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) )
357	356	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) )
358		opeq2	⊢ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 → ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ = ⟨ ( 𝑀 + 1 ) , 0 ⟩ )
359	358	sneqd	⊢ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 → { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } = { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } )
360		uneq12	⊢ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∧ { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } = { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) → ( ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
361	357 359 360	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ) → ( ( ( 1^st ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
362	355 361	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ) → ( 1^st ‘ 𝑘 ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
363	362	adantrr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ( 1^st ‘ 𝑘 ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
364		fnsnsplit	⊢ ( ( ( 2^nd ‘ 𝑘 ) Fn ( 1 ... ( 𝑀 + 1 ) ) ∧ ( 𝑀 + 1 ) ∈ ( 1 ... ( 𝑀 + 1 ) ) ) → ( 2^nd ‘ 𝑘 ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) )
365	92 96 364	syl2anr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( 2^nd ‘ 𝑘 ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) )
366	365	adantr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( 2^nd ‘ 𝑘 ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) )
367	125	reseq2d	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) )
368	367	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) )
369		opeq2	⊢ ( ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) → ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ = ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ )
370	369	sneqd	⊢ ( ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) → { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } = { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } )
371		uneq12	⊢ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) = ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∧ { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } = { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
372	368 370 371	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ( ( 2^nd ‘ 𝑘 ) ↾ ( ( 1 ... ( 𝑀 + 1 ) ) ∖ { ( 𝑀 + 1 ) } ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) ⟩ } ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
373	366 372	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( 2^nd ‘ 𝑘 ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
374	373	adantrl	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ( 2^nd ‘ 𝑘 ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
375	363 374	opeq12d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ⟨ ( 1^st ‘ 𝑘 ) , ( 2^nd ‘ 𝑘 ) ⟩ = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
376	352 375	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → 𝑘 = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
377	376	3adantr1	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → 𝑘 = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
378		fvex	⊢ ( 1^st ‘ 𝑘 ) ∈ V
379	378	resex	⊢ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∈ V
380	379 132	op1std	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( 1^st ‘ 𝑡 ) = ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) )
381	379 132	op2ndd	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( 2^nd ‘ 𝑡 ) = ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) )
382	381	imaeq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) )
383	382	xpeq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
384	381	imaeq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) )
385	384	xpeq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) )
386	383 385	uneq12d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) )
387	380 386	oveq12d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) )
388	387	uneq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) )
389	388	csbeq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
390	389	eqeq2d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
391	390	rexbidv	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
392	391	ralbidv	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
393	380	uneq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
394	381	uneq1d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
395	393 394	opeq12d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
396	395	eqeq2d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ 𝑘 = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
397	392 396	anbi12d	⊢ ( 𝑡 = ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ → ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ↔ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ) )
398	397	rspcev	⊢ ( ( ⟨ ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) , ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( ( 1^st ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( ( 2^nd ‘ 𝑘 ) ↾ ( 1 ... 𝑀 ) ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ) → ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
399	137 350 377 398	syl12anc	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) ∧ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) → ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
400	399	ex	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ) )
401		elrabi	⊢ ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } → 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) )
402		elrabi	⊢ ( 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } → 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) )
403	401 402	anim12i	⊢ ( ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∧ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) → ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) )
404		eqtr2	⊢ ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
405	22 24	opth	⊢ ( ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) )
406		difeq1	⊢ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) → ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
407		difun2	⊢ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } )
408		difun2	⊢ ( ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } )
409	406 407 408	3eqtr3g	⊢ ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) → ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
410		difeq1	⊢ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) → ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
411		difun2	⊢ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } )
412		difun2	⊢ ( ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } )
413	410 411 412	3eqtr3g	⊢ ( ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) → ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
414	409 413	anim12i	⊢ ( ( ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) → ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) )
415	405 414	sylbi	⊢ ( ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ → ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) )
416	404 415	syl	⊢ ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) )
417		elmapfn	⊢ ( ( 1^st ‘ 𝑡 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) → ( 1^st ‘ 𝑡 ) Fn ( 1 ... 𝑀 ) )
418		fnop	⊢ ( ( ( 1^st ‘ 𝑡 ) Fn ( 1 ... 𝑀 ) ∧ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑡 ) ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) )
419	418	ex	⊢ ( ( 1^st ‘ 𝑡 ) Fn ( 1 ... 𝑀 ) → ( ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑡 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
420	9 417 419	3syl	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑡 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
421	420 122	nsyli	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 𝜑 → ¬ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑡 ) ) )
422	421	impcom	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ¬ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑡 ) )
423		difsn	⊢ ( ¬ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑡 ) → ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( 1^st ‘ 𝑡 ) )
424	422 423	syl	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( 1^st ‘ 𝑡 ) )
425		xp1st	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 1^st ‘ 𝑛 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) )
426		elmapfn	⊢ ( ( 1^st ‘ 𝑛 ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) → ( 1^st ‘ 𝑛 ) Fn ( 1 ... 𝑀 ) )
427		fnop	⊢ ( ( ( 1^st ‘ 𝑛 ) Fn ( 1 ... 𝑀 ) ∧ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑛 ) ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) )
428	427	ex	⊢ ( ( 1^st ‘ 𝑛 ) Fn ( 1 ... 𝑀 ) → ( ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑛 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
429	425 426 428	3syl	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑛 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
430	429 122	nsyli	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 𝜑 → ¬ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑛 ) ) )
431	430	impcom	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ¬ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑛 ) )
432		difsn	⊢ ( ¬ ⟨ ( 𝑀 + 1 ) , 0 ⟩ ∈ ( 1^st ‘ 𝑛 ) → ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( 1^st ‘ 𝑛 ) )
433	431 432	syl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( 1^st ‘ 𝑛 ) )
434	424 433	eqeqan12d	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ∧ ( 𝜑 ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ↔ ( 1^st ‘ 𝑡 ) = ( 1^st ‘ 𝑛 ) ) )
435	434	anandis	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ↔ ( 1^st ‘ 𝑡 ) = ( 1^st ‘ 𝑛 ) ) )
436		f1ofn	⊢ ( ( 2^nd ‘ 𝑡 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) → ( 2^nd ‘ 𝑡 ) Fn ( 1 ... 𝑀 ) )
437		fnop	⊢ ( ( ( 2^nd ‘ 𝑡 ) Fn ( 1 ... 𝑀 ) ∧ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑡 ) ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) )
438	437	ex	⊢ ( ( 2^nd ‘ 𝑡 ) Fn ( 1 ... 𝑀 ) → ( ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑡 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
439	17 436 438	3syl	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑡 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
440	439 122	nsyli	⊢ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 𝜑 → ¬ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑡 ) ) )
441	440	impcom	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ¬ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑡 ) )
442		difsn	⊢ ( ¬ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑡 ) → ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( 2^nd ‘ 𝑡 ) )
443	441 442	syl	⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( 2^nd ‘ 𝑡 ) )
444		xp2nd	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 2^nd ‘ 𝑛 ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } )
445		fvex	⊢ ( 2^nd ‘ 𝑛 ) ∈ V
446		f1oeq1	⊢ ( 𝑓 = ( 2^nd ‘ 𝑛 ) → ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ↔ ( 2^nd ‘ 𝑛 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) ) )
447	445 446	elab	⊢ ( ( 2^nd ‘ 𝑛 ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ↔ ( 2^nd ‘ 𝑛 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
448	444 447	sylib	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 2^nd ‘ 𝑛 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) )
449		f1ofn	⊢ ( ( 2^nd ‘ 𝑛 ) : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) → ( 2^nd ‘ 𝑛 ) Fn ( 1 ... 𝑀 ) )
450		fnop	⊢ ( ( ( 2^nd ‘ 𝑛 ) Fn ( 1 ... 𝑀 ) ∧ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑛 ) ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) )
451	450	ex	⊢ ( ( 2^nd ‘ 𝑛 ) Fn ( 1 ... 𝑀 ) → ( ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑛 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
452	448 449 451	3syl	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑛 ) → ( 𝑀 + 1 ) ∈ ( 1 ... 𝑀 ) ) )
453	452 122	nsyli	⊢ ( 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) → ( 𝜑 → ¬ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑛 ) ) )
454	453	impcom	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ¬ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑛 ) )
455		difsn	⊢ ( ¬ ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ ∈ ( 2^nd ‘ 𝑛 ) → ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( 2^nd ‘ 𝑛 ) )
456	454 455	syl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( 2^nd ‘ 𝑛 ) )
457	443 456	eqeqan12d	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ∧ ( 𝜑 ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ↔ ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑛 ) ) )
458	457	anandis	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ↔ ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑛 ) ) )
459	435 458	anbi12d	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) ↔ ( ( 1^st ‘ 𝑡 ) = ( 1^st ‘ 𝑛 ) ∧ ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑛 ) ) ) )
460		xpopth	⊢ ( ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) → ( ( ( 1^st ‘ 𝑡 ) = ( 1^st ‘ 𝑛 ) ∧ ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑛 ) ) ↔ 𝑡 = 𝑛 ) )
461	460	adantl	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( 1^st ‘ 𝑡 ) = ( 1^st ‘ 𝑛 ) ∧ ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑛 ) ) ↔ 𝑡 = 𝑛 ) )
462	459 461	bitrd	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( ( ( 1^st ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) ∧ ( ( 2^nd ‘ 𝑡 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∖ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ) ↔ 𝑡 = 𝑛 ) )
463	416 462	imbitrid	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∧ 𝑛 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ) ) → ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) )
464	403 463	sylan2	⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∧ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) → ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) )
465	464	ralrimivva	⊢ ( 𝜑 → ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) )
466	465	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) )
467	400 466	jctird	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ( ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ∧ ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) ) ) )
468		fveq2	⊢ ( 𝑡 = 𝑛 → ( 1^st ‘ 𝑡 ) = ( 1^st ‘ 𝑛 ) )
469	468	uneq1d	⊢ ( 𝑡 = 𝑛 → ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) = ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) )
470		fveq2	⊢ ( 𝑡 = 𝑛 → ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑛 ) )
471	470	uneq1d	⊢ ( 𝑡 = 𝑛 → ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) = ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) )
472	469 471	opeq12d	⊢ ( 𝑡 = 𝑛 → ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
473	472	eqeq2d	⊢ ( 𝑡 = 𝑛 → ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
474	473	reu4	⊢ ( ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ ( ∃ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) ) )
475	58	rexrab	⊢ ( ∃ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
476	475	anbi1i	⊢ ( ( ∃ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) ) ↔ ( ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ∧ ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) ) )
477	474 476	bitri	⊢ ( ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ ( ∃ 𝑡 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑡 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑡 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑡 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) ∧ ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∀ 𝑛 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ( ( 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∧ 𝑘 = ⟨ ( ( 1^st ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑛 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) → 𝑡 = 𝑛 ) ) )
478	467 477	imbitrrdi	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ) → ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
479	478	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
480		fveq2	⊢ ( 𝑠 = 𝑘 → ( 1^st ‘ 𝑠 ) = ( 1^st ‘ 𝑘 ) )
481		fveq2	⊢ ( 𝑠 = 𝑘 → ( 2^nd ‘ 𝑠 ) = ( 2^nd ‘ 𝑘 ) )
482	481	imaeq1d	⊢ ( 𝑠 = 𝑘 → ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) )
483	482	xpeq1d	⊢ ( 𝑠 = 𝑘 → ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
484	481	imaeq1d	⊢ ( 𝑠 = 𝑘 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) = ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) )
485	484	xpeq1d	⊢ ( 𝑠 = 𝑘 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) )
486	483 485	uneq12d	⊢ ( 𝑠 = 𝑘 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) )
487	480 486	oveq12d	⊢ ( 𝑠 = 𝑘 → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) )
488	487	uneq1d	⊢ ( 𝑠 = 𝑘 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
489	488	csbeq1d	⊢ ( 𝑠 = 𝑘 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
490	489	eqeq2d	⊢ ( 𝑠 = 𝑘 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
491	490	rexbidv	⊢ ( 𝑠 = 𝑘 → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
492	491	ralbidv	⊢ ( 𝑠 = 𝑘 → ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
493	480	fveq1d	⊢ ( 𝑠 = 𝑘 → ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) )
494	493	eqeq1d	⊢ ( 𝑠 = 𝑘 → ( ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ↔ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ) )
495	481	fveq1d	⊢ ( 𝑠 = 𝑘 → ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) )
496	495	eqeq1d	⊢ ( 𝑠 = 𝑘 → ( ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ↔ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) )
497	492 494 496	3anbi123d	⊢ ( 𝑠 = 𝑘 → ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ↔ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) ) )
498	497	ralrab	⊢ ( ∀ 𝑘 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ↔ ∀ 𝑘 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ( ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑘 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑘 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑘 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑘 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) → ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
499	479 498	sylibr	⊢ ( 𝜑 → ∀ 𝑘 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
500		eqid	⊢ ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ↦ ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) = ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ↦ ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ )
501	500	f1ompt	⊢ ( ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ↦ ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) : { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } –1-1-onto→ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ↔ ( ∀ 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ∧ ∀ 𝑘 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } ∃! 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } 𝑘 = ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) )
502	60 499 501	sylanbrc	⊢ ( 𝜑 → ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ↦ ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) : { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } –1-1-onto→ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } )
503		ovex	⊢ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) ∈ V
504		ovex	⊢ ( ( 1 ... 𝑀 ) ↑_m ( 1 ... 𝑀 ) ) ∈ V
505		f1of	⊢ ( 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) → 𝑓 : ( 1 ... 𝑀 ) ⟶ ( 1 ... 𝑀 ) )
506	505	ss2abi	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ⊆ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) ⟶ ( 1 ... 𝑀 ) }
507	68 68	mapval	⊢ ( ( 1 ... 𝑀 ) ↑_m ( 1 ... 𝑀 ) ) = { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) ⟶ ( 1 ... 𝑀 ) }
508	506 507	sseqtrri	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ⊆ ( ( 1 ... 𝑀 ) ↑_m ( 1 ... 𝑀 ) )
509	504 508	ssexi	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ∈ V
510	503 509	xpex	⊢ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∈ V
511	510	rabex	⊢ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ V
512	511	f1oen	⊢ ( ( 𝑡 ∈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ↦ ⟨ ( ( 1^st ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , 0 ⟩ } ) , ( ( 2^nd ‘ 𝑡 ) ∪ { ⟨ ( 𝑀 + 1 ) , ( 𝑀 + 1 ) ⟩ } ) ⟩ ) : { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } –1-1-onto→ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ≈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } )
513	502 512	syl	⊢ ( 𝜑 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑀 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑀 ) –1-1-onto→ ( 1 ... 𝑀 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑀 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑀 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ≈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑀 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑀 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑀 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ∃ 𝑗 ∈ ( 0 ... 𝑀 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑀 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑀 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑀 + 1 ) ) = ( 𝑀 + 1 ) ) } )

Metamath Proof Explorer

Theorem poimirlem4

Proof