poimirlem15

	Ref	Expression
Hypotheses	poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	poimirlem22.s	⊢ 𝑆 = { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) }
	poimirlem22.1	⊢ ( 𝜑 → 𝐹 : ( 0 ... ( 𝑁 − 1 ) ) ⟶ ( ( 0 ... 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
	poimirlem22.2	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
	poimirlem15.3	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) )
Assertion	poimirlem15	⊢ ( 𝜑 → ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		poimirlem22.s	⊢ 𝑆 = { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) }
3		poimirlem22.1	⊢ ( 𝜑 → 𝐹 : ( 0 ... ( 𝑁 − 1 ) ) ⟶ ( ( 0 ... 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
4		poimirlem22.2	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
5		poimirlem15.3	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) )
6		elrabi	⊢ ( 𝑇 ∈ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) } → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
7	6 2	eleq2s	⊢ ( 𝑇 ∈ 𝑆 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
8	4 7	syl	⊢ ( 𝜑 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
9		xp1st	⊢ ( 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) → ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
10		xp1st	⊢ ( ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
11	8 9 10	3syl	⊢ ( 𝜑 → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
12		xp2nd	⊢ ( ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
13	8 9 12	3syl	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
14		fvex	⊢ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ V
15		f1oeq1	⊢ ( 𝑓 = ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
16	14 15	elab	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ↔ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
17	13 16	sylib	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
18		elfznn	⊢ ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( 2^nd ‘ 𝑇 ) ∈ ℕ )
19	5 18	syl	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ℕ )
20	19	nnred	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ℝ )
21	20	ltp1d	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) < ( ( 2^nd ‘ 𝑇 ) + 1 ) )
22	20 21	ltned	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ≠ ( ( 2^nd ‘ 𝑇 ) + 1 ) )
23	22	necomd	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) + 1 ) ≠ ( 2^nd ‘ 𝑇 ) )
24		fvex	⊢ ( 2^nd ‘ 𝑇 ) ∈ V
25		ovex	⊢ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ V
26		f1oprg	⊢ ( ( ( ( 2^nd ‘ 𝑇 ) ∈ V ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ V ) ∧ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ V ∧ ( 2^nd ‘ 𝑇 ) ∈ V ) ) → ( ( ( 2^nd ‘ 𝑇 ) ≠ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≠ ( 2^nd ‘ 𝑇 ) ) → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } ) )
27	24 25 25 24 26	mp4an	⊢ ( ( ( 2^nd ‘ 𝑇 ) ≠ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≠ ( 2^nd ‘ 𝑇 ) ) → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } )
28	22 23 27	syl2anc	⊢ ( 𝜑 → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } )
29		prcom	⊢ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) }
30		f1oeq3	⊢ ( { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } ↔ { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
31	29 30	ax-mp	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } ↔ { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
32	28 31	sylib	⊢ ( 𝜑 → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
33		f1oi	⊢ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) : ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) –1-1-onto→ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
34		disjdif	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ∅
35		f1oun	⊢ ( ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∧ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) : ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) –1-1-onto→ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ∧ ( ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ∅ ∧ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ∅ ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) –1-1-onto→ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
36	34 34 35	mpanr12	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∧ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) : ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) –1-1-onto→ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) –1-1-onto→ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
37	32 33 36	sylancl	⊢ ( 𝜑 → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) –1-1-onto→ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
38	1	nncnd	⊢ ( 𝜑 → 𝑁 ∈ ℂ )
39		npcan1	⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
40	38 39	syl	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
41	1	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
42		peano2zm	⊢ ( 𝑁 ∈ ℤ → ( 𝑁 − 1 ) ∈ ℤ )
43	41 42	syl	⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℤ )
44		uzid	⊢ ( ( 𝑁 − 1 ) ∈ ℤ → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
45		peano2uz	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
46	43 44 45	3syl	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
47	40 46	eqeltrrd	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
48		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... 𝑁 ) )
49	47 48	syl	⊢ ( 𝜑 → ( 1 ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... 𝑁 ) )
50	49 5	sseldd	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... 𝑁 ) )
51	19	peano2nnd	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ )
52	43	zred	⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℝ )
53	1	nnred	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
54		elfzle2	⊢ ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( 2^nd ‘ 𝑇 ) ≤ ( 𝑁 − 1 ) )
55	5 54	syl	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ≤ ( 𝑁 − 1 ) )
56	53	ltm1d	⊢ ( 𝜑 → ( 𝑁 − 1 ) < 𝑁 )
57	20 52 53 55 56	lelttrd	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) < 𝑁 )
58	19	nnzd	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ℤ )
59		zltp1le	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 2^nd ‘ 𝑇 ) < 𝑁 ↔ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) )
60	58 41 59	syl2anc	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) < 𝑁 ↔ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) )
61	57 60	mpbid	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 )
62		fznn	⊢ ( 𝑁 ∈ ℤ → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑁 ) ↔ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) ) )
63	41 62	syl	⊢ ( 𝜑 → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑁 ) ↔ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) ) )
64	51 61 63	mpbir2and	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑁 ) )
65		prssi	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... 𝑁 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑁 ) ) → { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... 𝑁 ) )
66	50 64 65	syl2anc	⊢ ( 𝜑 → { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... 𝑁 ) )
67		undif	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... 𝑁 ) ↔ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( 1 ... 𝑁 ) )
68	66 67	sylib	⊢ ( 𝜑 → ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( 1 ... 𝑁 ) )
69		f1oeq23	⊢ ( ( ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( 1 ... 𝑁 ) ∧ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( 1 ... 𝑁 ) ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) –1-1-onto→ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ↔ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
70	68 68 69	syl2anc	⊢ ( 𝜑 → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) –1-1-onto→ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ↔ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
71	37 70	mpbid	⊢ ( 𝜑 → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
72		f1oco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ∧ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
73	17 71 72	syl2anc	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
74		prex	⊢ { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∈ V
75		ovex	⊢ ( 1 ... 𝑁 ) ∈ V
76		difexg	⊢ ( ( 1 ... 𝑁 ) ∈ V → ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∈ V )
77		resiexg	⊢ ( ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∈ V → ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ∈ V )
78	75 76 77	mp2b	⊢ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ∈ V
79	74 78	unex	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ∈ V
80	14 79	coex	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ∈ V
81		f1oeq1	⊢ ( 𝑓 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) → ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
82	80 81	elab	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ↔ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
83	73 82	sylibr	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
84		opelxpi	⊢ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
85	11 83 84	syl2anc	⊢ ( 𝜑 → ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
86		fz1ssfz0	⊢ ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 )
87	49 86	sstrdi	⊢ ( 𝜑 → ( 1 ... ( 𝑁 − 1 ) ) ⊆ ( 0 ... 𝑁 ) )
88	87 5	sseldd	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ( 0 ... 𝑁 ) )
89		opelxpi	⊢ ( ( ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ( 2^nd ‘ 𝑇 ) ∈ ( 0 ... 𝑁 ) ) → ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
90	85 88 89	syl2anc	⊢ ( 𝜑 → ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
91		fveq2	⊢ ( 𝑡 = 𝑇 → ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) )
92	91	breq2d	⊢ ( 𝑡 = 𝑇 → ( 𝑦 < ( 2^nd ‘ 𝑡 ) ↔ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
93	92	ifbid	⊢ ( 𝑡 = 𝑇 → if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) )
94	93	csbeq1d	⊢ ( 𝑡 = 𝑇 → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
95		2fveq3	⊢ ( 𝑡 = 𝑇 → ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) )
96		2fveq3	⊢ ( 𝑡 = 𝑇 → ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) )
97	96	imaeq1d	⊢ ( 𝑡 = 𝑇 → ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) )
98	97	xpeq1d	⊢ ( 𝑡 = 𝑇 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
99	96	imaeq1d	⊢ ( 𝑡 = 𝑇 → ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) )
100	99	xpeq1d	⊢ ( 𝑡 = 𝑇 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) )
101	98 100	uneq12d	⊢ ( 𝑡 = 𝑇 → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
102	95 101	oveq12d	⊢ ( 𝑡 = 𝑇 → ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
103	102	csbeq2dv	⊢ ( 𝑡 = 𝑇 → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
104	94 103	eqtrd	⊢ ( 𝑡 = 𝑇 → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
105	104	mpteq2dv	⊢ ( 𝑡 = 𝑇 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
106	105	eqeq2d	⊢ ( 𝑡 = 𝑇 → ( 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ↔ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) )
107	106 2	elrab2	⊢ ( 𝑇 ∈ 𝑆 ↔ ( 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∧ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) )
108	107	simprbi	⊢ ( 𝑇 ∈ 𝑆 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
109	4 108	syl	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
110		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... 𝑦 ) ) )
111		f1ofn	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
112	28 111	syl	⊢ ( 𝜑 → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
113	112	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
114		incom	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( 1 ... 𝑦 ) ) = ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
115		elfznn0	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℕ₀ )
116	115	nn0red	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℝ )
117		ltnle	⊢ ( ( 𝑦 ∈ ℝ ∧ ( 2^nd ‘ 𝑇 ) ∈ ℝ ) → ( 𝑦 < ( 2^nd ‘ 𝑇 ) ↔ ¬ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) )
118	116 20 117	syl2anr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 < ( 2^nd ‘ 𝑇 ) ↔ ¬ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) )
119	118	biimpa	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ¬ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 )
120		elfzle2	⊢ ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... 𝑦 ) → ( 2^nd ‘ 𝑇 ) ≤ 𝑦 )
121	119 120	nsyl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ¬ ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... 𝑦 ) )
122		disjsn	⊢ ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) = ∅ ↔ ¬ ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... 𝑦 ) )
123	121 122	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) = ∅ )
124	116	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → 𝑦 ∈ ℝ )
125	20	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 2^nd ‘ 𝑇 ) ∈ ℝ )
126	51	nnred	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℝ )
127	126	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℝ )
128		simpr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → 𝑦 < ( 2^nd ‘ 𝑇 ) )
129	21	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 2^nd ‘ 𝑇 ) < ( ( 2^nd ‘ 𝑇 ) + 1 ) )
130	124 125 127 128 129	lttrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → 𝑦 < ( ( 2^nd ‘ 𝑇 ) + 1 ) )
131		ltnle	⊢ ( ( 𝑦 ∈ ℝ ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℝ ) → ( 𝑦 < ( ( 2^nd ‘ 𝑇 ) + 1 ) ↔ ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑦 ) )
132	116 126 131	syl2anr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 < ( ( 2^nd ‘ 𝑇 ) + 1 ) ↔ ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑦 ) )
133	132	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 𝑦 < ( ( 2^nd ‘ 𝑇 ) + 1 ) ↔ ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑦 ) )
134	130 133	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑦 )
135		elfzle2	⊢ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑦 )
136	134 135	nsyl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑦 ) )
137		disjsn	⊢ ( ( ( 1 ... 𝑦 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... 𝑦 ) )
138	136 137	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 1 ... 𝑦 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
139	123 138	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) ∪ ( ( 1 ... 𝑦 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ∅ ∪ ∅ ) )
140		df-pr	⊢ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } = ( { ( 2^nd ‘ 𝑇 ) } ∪ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
141	140	ineq2i	⊢ ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ( ( 1 ... 𝑦 ) ∩ ( { ( 2^nd ‘ 𝑇 ) } ∪ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
142		indi	⊢ ( ( 1 ... 𝑦 ) ∩ ( { ( 2^nd ‘ 𝑇 ) } ∪ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) ∪ ( ( 1 ... 𝑦 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
143	141 142	eqtr2i	⊢ ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) ∪ ( ( 1 ... 𝑦 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
144		un0	⊢ ( ∅ ∪ ∅ ) = ∅
145	139 143 144	3eqtr3g	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
146	114 145	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( 1 ... 𝑦 ) ) = ∅ )
147		fnimadisj	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∧ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( 1 ... 𝑦 ) ) = ∅ ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... 𝑦 ) ) = ∅ )
148	113 146 147	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... 𝑦 ) ) = ∅ )
149	40	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
150		elfzuz3	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
151		peano2uz	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
152	150 151	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
153	152	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
154	149 153	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) )
155		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) → ( 1 ... 𝑦 ) ⊆ ( 1 ... 𝑁 ) )
156		reldisj	⊢ ( ( 1 ... 𝑦 ) ⊆ ( 1 ... 𝑁 ) → ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ( 1 ... 𝑦 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
157	154 155 156	3syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ( 1 ... 𝑦 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
158	157	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( 1 ... 𝑦 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ( 1 ... 𝑦 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
159	145 158	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 1 ... 𝑦 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
160		resiima	⊢ ( ( 1 ... 𝑦 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... 𝑦 ) ) = ( 1 ... 𝑦 ) )
161	159 160	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... 𝑦 ) ) = ( 1 ... 𝑦 ) )
162	148 161	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... 𝑦 ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... 𝑦 ) ) ) = ( ∅ ∪ ( 1 ... 𝑦 ) ) )
163		imaundir	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... 𝑦 ) ) = ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... 𝑦 ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... 𝑦 ) ) )
164		uncom	⊢ ( ∅ ∪ ( 1 ... 𝑦 ) ) = ( ( 1 ... 𝑦 ) ∪ ∅ )
165		un0	⊢ ( ( 1 ... 𝑦 ) ∪ ∅ ) = ( 1 ... 𝑦 )
166	164 165	eqtr2i	⊢ ( 1 ... 𝑦 ) = ( ∅ ∪ ( 1 ... 𝑦 ) )
167	162 163 166	3eqtr4g	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... 𝑦 ) ) = ( 1 ... 𝑦 ) )
168	167	imaeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... 𝑦 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) )
169	110 168	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) )
170	169	xpeq1d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) )
171		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
172		imaundir	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
173		imassrn	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⊆ ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ }
174	173	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⊆ ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
175		fnima	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
176	28 111 175	3syl	⊢ ( 𝜑 → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
177	176	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
178		elfzelz	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ )
179		zltp1le	⊢ ( ( 𝑦 ∈ ℤ ∧ ( 2^nd ‘ 𝑇 ) ∈ ℤ ) → ( 𝑦 < ( 2^nd ‘ 𝑇 ) ↔ ( 𝑦 + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ) )
180	178 58 179	syl2anr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 < ( 2^nd ‘ 𝑇 ) ↔ ( 𝑦 + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ) )
181	180	biimpa	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 𝑦 + 1 ) ≤ ( 2^nd ‘ 𝑇 ) )
182	20 53 57	ltled	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ≤ 𝑁 )
183	182	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 2^nd ‘ 𝑇 ) ≤ 𝑁 )
184	58	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2^nd ‘ 𝑇 ) ∈ ℤ )
185		nn0p1nn	⊢ ( 𝑦 ∈ ℕ₀ → ( 𝑦 + 1 ) ∈ ℕ )
186	115 185	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℕ )
187	186	nnzd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℤ )
188	187	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ℤ )
189	41	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ℤ )
190		elfz	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 2^nd ‘ 𝑇 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( ( 𝑦 + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑁 ) ) )
191	184 188 189 190	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ 𝑇 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( ( 𝑦 + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑁 ) ) )
192	191	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 2^nd ‘ 𝑇 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( ( 𝑦 + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑁 ) ) )
193	181 183 192	mpbir2and	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 2^nd ‘ 𝑇 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) )
194		1red	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → 1 ∈ ℝ )
195		ltle	⊢ ( ( 𝑦 ∈ ℝ ∧ ( 2^nd ‘ 𝑇 ) ∈ ℝ ) → ( 𝑦 < ( 2^nd ‘ 𝑇 ) → 𝑦 ≤ ( 2^nd ‘ 𝑇 ) ) )
196	116 20 195	syl2anr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 < ( 2^nd ‘ 𝑇 ) → 𝑦 ≤ ( 2^nd ‘ 𝑇 ) ) )
197	196	imp	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → 𝑦 ≤ ( 2^nd ‘ 𝑇 ) )
198	124 125 194 197	leadd1dd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 𝑦 + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) )
199	61	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 )
200	58	peano2zd	⊢ ( 𝜑 → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℤ )
201	200	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℤ )
202		elfz	⊢ ( ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( ( 𝑦 + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) ) )
203	201 188 189 202	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( ( 𝑦 + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) ) )
204	203	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( ( 𝑦 + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ 𝑁 ) ) )
205	198 199 204	mpbir2and	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) )
206		prssi	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( ( 𝑦 + 1 ) ... 𝑁 ) )
207	193 205 206	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( ( 𝑦 + 1 ) ... 𝑁 ) )
208		imass2	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( ( 𝑦 + 1 ) ... 𝑁 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
209	207 208	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
210	177 209	eqsstrrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ⊆ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
211	174 210	eqssd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
212		f1ofo	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –1-1-onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } )
213		forn	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } : { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } –onto→ { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } = { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } )
214	28 212 213	3syl	⊢ ( 𝜑 → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } = { ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) } )
215	214 29	eqtrdi	⊢ ( 𝜑 → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
216	215	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
217	211 216	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
218		undif	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( ( 𝑦 + 1 ) ... 𝑁 ) ↔ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( 𝑦 + 1 ) ... 𝑁 ) )
219	207 218	sylib	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( 𝑦 + 1 ) ... 𝑁 ) )
220	219	imaeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
221		fnresi	⊢ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
222		disjdifr	⊢ ( ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅
223		fnimadisj	⊢ ( ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∧ ( ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
224	221 222 223	mp2an	⊢ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅
225	224	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
226		nnuz	⊢ ℕ = ( ℤ_≥ ‘ 1 )
227	186 226	eleqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
228		fzss1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
229	227 228	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
230	229	ssdifd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
231		resiima	⊢ ( ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
232	230 231	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
233	232	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
234	225 233	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ∅ ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
235		imaundi	⊢ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
236		uncom	⊢ ( ∅ ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ∅ )
237		un0	⊢ ( ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ∅ ) = ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
238	236 237	eqtr2i	⊢ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ( ∅ ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
239	234 235 238	3eqtr4g	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
240	220 239	eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
241	217 240	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
242	172 241	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( ( 𝑦 + 1 ) ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
243	242 219	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( 𝑦 + 1 ) ... 𝑁 ) )
244	243	imaeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
245	171 244	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
246	245	xpeq1d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) )
247	170 246	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
248	247	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
249		iftrue	⊢ ( 𝑦 < ( 2^nd ‘ 𝑇 ) → if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = 𝑦 )
250	249	csbeq1d	⊢ ( 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑦 / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
251		vex	⊢ 𝑦 ∈ V
252		oveq2	⊢ ( 𝑗 = 𝑦 → ( 1 ... 𝑗 ) = ( 1 ... 𝑦 ) )
253	252	imaeq2d	⊢ ( 𝑗 = 𝑦 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) )
254	253	xpeq1d	⊢ ( 𝑗 = 𝑦 → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) )
255		oveq1	⊢ ( 𝑗 = 𝑦 → ( 𝑗 + 1 ) = ( 𝑦 + 1 ) )
256	255	oveq1d	⊢ ( 𝑗 = 𝑦 → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( 𝑦 + 1 ) ... 𝑁 ) )
257	256	imaeq2d	⊢ ( 𝑗 = 𝑦 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
258	257	xpeq1d	⊢ ( 𝑗 = 𝑦 → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) )
259	254 258	uneq12d	⊢ ( 𝑗 = 𝑦 → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
260	259	oveq2d	⊢ ( 𝑗 = 𝑦 → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
261	251 260	csbie	⊢ ⦋ 𝑦 / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
262	250 261	eqtrdi	⊢ ( 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
263	262	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
264	249	csbeq1d	⊢ ( 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑦 / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
265	252	imaeq2d	⊢ ( 𝑗 = 𝑦 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) )
266	265	xpeq1d	⊢ ( 𝑗 = 𝑦 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) )
267	256	imaeq2d	⊢ ( 𝑗 = 𝑦 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
268	267	xpeq1d	⊢ ( 𝑗 = 𝑦 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) )
269	266 268	uneq12d	⊢ ( 𝑗 = 𝑦 → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
270	269	oveq2d	⊢ ( 𝑗 = 𝑦 → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
271	251 270	csbie	⊢ ⦋ 𝑦 / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
272	264 271	eqtrdi	⊢ ( 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
273	272	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
274	248 263 273	3eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
275		lenlt	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ↔ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
276	20 116 275	syl2an	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ↔ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
277	276	biimpar	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( 2^nd ‘ 𝑇 ) ≤ 𝑦 )
278		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
279		imaundir	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
280		imassrn	⊢ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ }
281	280	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
282	176	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
283	19	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) ∈ ℕ )
284	20	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) ∈ ℝ )
285	116	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → 𝑦 ∈ ℝ )
286	186	nnred	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℝ )
287	286	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 𝑦 + 1 ) ∈ ℝ )
288		simpr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) ≤ 𝑦 )
289	116	lep1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ≤ ( 𝑦 + 1 ) )
290	289	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → 𝑦 ≤ ( 𝑦 + 1 ) )
291	284 285 287 288 290	letrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) ≤ ( 𝑦 + 1 ) )
292		fznn	⊢ ( ( 𝑦 + 1 ) ∈ ℤ → ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( ( 2^nd ‘ 𝑇 ) ∈ ℕ ∧ ( 2^nd ‘ 𝑇 ) ≤ ( 𝑦 + 1 ) ) ) )
293	187 292	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( ( 2^nd ‘ 𝑇 ) ∈ ℕ ∧ ( 2^nd ‘ 𝑇 ) ≤ ( 𝑦 + 1 ) ) ) )
294	293	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( ( 2^nd ‘ 𝑇 ) ∈ ℕ ∧ ( 2^nd ‘ 𝑇 ) ≤ ( 𝑦 + 1 ) ) ) )
295	283 291 294	mpbir2and	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) )
296	51	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ )
297		1red	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → 1 ∈ ℝ )
298	284 285 297 288	leadd1dd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ ( 𝑦 + 1 ) )
299		fznn	⊢ ( ( 𝑦 + 1 ) ∈ ℤ → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ ( 𝑦 + 1 ) ) ) )
300	187 299	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ ( 𝑦 + 1 ) ) ) )
301	300	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℕ ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ≤ ( 𝑦 + 1 ) ) ) )
302	296 298 301	mpbir2and	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) )
303		prssi	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ∧ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... ( 𝑦 + 1 ) ) )
304	295 302 303	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... ( 𝑦 + 1 ) ) )
305		imass2	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... ( 𝑦 + 1 ) ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) )
306	304 305	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) )
307	282 306	eqsstrrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ⊆ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) )
308	281 307	eqssd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) = ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } )
309	215	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ran { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
310	308 309	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) = { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
311		undif	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ⊆ ( 1 ... ( 𝑦 + 1 ) ) ↔ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( 1 ... ( 𝑦 + 1 ) ) )
312	304 311	sylib	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( 1 ... ( 𝑦 + 1 ) ) )
313	312	imaeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
314	224	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
315		eluzp1p1	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
316	150 315	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
317	316	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
318	149 317	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
319		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
320	318 319	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
321	320	ssdifd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
322	321	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
323		resiima	⊢ ( ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
324	322 323	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
325	314 324	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ∅ ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
326		imaundi	⊢ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
327		uncom	⊢ ( ∅ ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ∅ )
328		un0	⊢ ( ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ∪ ∅ ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
329	327 328	eqtr2i	⊢ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ( ∅ ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
330	325 326 329	3eqtr4g	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
331	313 330	eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
332	310 331	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) = ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
333	279 332	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∪ ( ( 1 ... ( 𝑦 + 1 ) ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
334	333 312	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( 1 ... ( 𝑦 + 1 ) ) )
335	334	imaeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
336	278 335	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
337	336	xpeq1d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) )
338		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
339	112	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
340		incom	⊢ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
341	126	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℝ )
342	186	peano2nnd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℕ )
343	342	nnred	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℝ )
344	343	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℝ )
345	21	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) < ( ( 2^nd ‘ 𝑇 ) + 1 ) )
346	116	ltp1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 < ( 𝑦 + 1 ) )
347	346	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → 𝑦 < ( 𝑦 + 1 ) )
348	284 285 287 288 347	lelttrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) < ( 𝑦 + 1 ) )
349	284 287 297 348	ltadd1dd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) + 1 ) < ( ( 𝑦 + 1 ) + 1 ) )
350	284 341 344 345 349	lttrd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( 2^nd ‘ 𝑇 ) < ( ( 𝑦 + 1 ) + 1 ) )
351		ltnle	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ℝ ∧ ( ( 𝑦 + 1 ) + 1 ) ∈ ℝ ) → ( ( 2^nd ‘ 𝑇 ) < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ) )
352	20 343 351	syl2an	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ 𝑇 ) < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ) )
353	352	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ 𝑇 ) < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( 2^nd ‘ 𝑇 ) ) )
354	350 353	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( 2^nd ‘ 𝑇 ) )
355		elfzle1	⊢ ( ( 2^nd ‘ 𝑇 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → ( ( 𝑦 + 1 ) + 1 ) ≤ ( 2^nd ‘ 𝑇 ) )
356	354 355	nsyl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ¬ ( 2^nd ‘ 𝑇 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
357		disjsn	⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) = ∅ ↔ ¬ ( 2^nd ‘ 𝑇 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
358	356 357	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) = ∅ )
359		ltnle	⊢ ( ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ℝ ∧ ( ( 𝑦 + 1 ) + 1 ) ∈ ℝ ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) ) )
360	126 343 359	syl2an	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) ) )
361	360	adantr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( 2^nd ‘ 𝑇 ) + 1 ) < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) ) )
362	349 361	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) )
363		elfzle1	⊢ ( ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → ( ( 𝑦 + 1 ) + 1 ) ≤ ( ( 2^nd ‘ 𝑇 ) + 1 ) )
364	362 363	nsyl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
365		disjsn	⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ¬ ( ( 2^nd ‘ 𝑇 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
366	364 365	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
367	358 366	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) ∪ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ∅ ∪ ∅ ) )
368	140	ineq2i	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ ( { ( 2^nd ‘ 𝑇 ) } ∪ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
369		indi	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ ( { ( 2^nd ‘ 𝑇 ) } ∪ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) ∪ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
370	368 369	eqtr2i	⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) } ) ∪ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) = ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } )
371	367 370 144	3eqtr3g	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ )
372	340 371	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
373		fnimadisj	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } Fn { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∧ ( { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
374	339 372 373	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
375	342 226	eleqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
376		fzss1	⊢ ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
377		reldisj	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) → ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
378	375 376 377	3syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
379	378	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∩ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) = ∅ ↔ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) )
380	371 379	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) )
381		resiima	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
382	380 381	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
383	374 382	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ∅ ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
384		imaundir	⊢ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ∪ ( ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
385		uncom	⊢ ( ∅ ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∪ ∅ )
386		un0	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∪ ∅ ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 )
387	385 386	eqtr2i	⊢ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) = ( ∅ ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
388	383 384 387	3eqtr4g	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
389	388	imaeq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
390	338 389	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
391	390	xpeq1d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
392	337 391	uneq12d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
393	277 392	syldan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
394	393	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
395		iffalse	⊢ ( ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) → if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = ( 𝑦 + 1 ) )
396	395	csbeq1d	⊢ ( ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ ( 𝑦 + 1 ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
397		ovex	⊢ ( 𝑦 + 1 ) ∈ V
398		oveq2	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( 1 ... 𝑗 ) = ( 1 ... ( 𝑦 + 1 ) ) )
399	398	imaeq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
400	399	xpeq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) )
401		oveq1	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( 𝑗 + 1 ) = ( ( 𝑦 + 1 ) + 1 ) )
402	401	oveq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
403	402	imaeq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
404	403	xpeq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
405	400 404	uneq12d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
406	405	oveq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
407	397 406	csbie	⊢ ⦋ ( 𝑦 + 1 ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
408	396 407	eqtrdi	⊢ ( ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
409	408	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
410	395	csbeq1d	⊢ ( ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ ( 𝑦 + 1 ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
411	398	imaeq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
412	411	xpeq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) )
413	402	imaeq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
414	413	xpeq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
415	412 414	uneq12d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
416	415	oveq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
417	397 416	csbie	⊢ ⦋ ( 𝑦 + 1 ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
418	410 417	eqtrdi	⊢ ( ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
419	418	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
420	394 409 419	3eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
421	274 420	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
422	421	mpteq2dva	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
423	109 422	eqtr4d	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
424		opex	⊢ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ ∈ V
425	424 24	op1std	⊢ ( 𝑡 = ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ → ( 1^st ‘ 𝑡 ) = ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ )
426	424 24	op2ndd	⊢ ( 𝑡 = ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ → ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) )
427		breq2	⊢ ( ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) → ( 𝑦 < ( 2^nd ‘ 𝑡 ) ↔ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
428	427	ifbid	⊢ ( ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) → if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) )
429	428	csbeq1d	⊢ ( ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
430		fvex	⊢ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ V
431	430 80	op1std	⊢ ( ( 1^st ‘ 𝑡 ) = ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ → ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) )
432	430 80	op2ndd	⊢ ( ( 1^st ‘ 𝑡 ) = ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ → ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) )
433		id	⊢ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) → ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) )
434		imaeq1	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) )
435	434	xpeq1d	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
436		imaeq1	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) )
437	436	xpeq1d	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) )
438	435 437	uneq12d	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
439	433 438	oveqan12d	⊢ ( ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∧ ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
440	431 432 439	syl2anc	⊢ ( ( 1^st ‘ 𝑡 ) = ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ → ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
441	440	csbeq2dv	⊢ ( ( 1^st ‘ 𝑡 ) = ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
442	429 441	sylan9eqr	⊢ ( ( ( 1^st ‘ 𝑡 ) = ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ ∧ ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
443	425 426 442	syl2anc	⊢ ( 𝑡 = ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
444	443	mpteq2dv	⊢ ( 𝑡 = ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
445	444	eqeq2d	⊢ ( 𝑡 = ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ → ( 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ↔ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) )
446	445 2	elrab2	⊢ ( ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ ∈ 𝑆 ↔ ( ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∧ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) )
447	90 423 446	sylanbrc	⊢ ( 𝜑 → ⟨ ⟨ ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) , ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( { ⟨ ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) ⟩ , ⟨ ( ( 2^nd ‘ 𝑇 ) + 1 ) , ( 2^nd ‘ 𝑇 ) ⟩ } ∪ ( I ↾ ( ( 1 ... 𝑁 ) ∖ { ( 2^nd ‘ 𝑇 ) , ( ( 2^nd ‘ 𝑇 ) + 1 ) } ) ) ) ) ⟩ , ( 2^nd ‘ 𝑇 ) ⟩ ∈ 𝑆 )

Metamath Proof Explorer

Theorem poimirlem15

Proof