Metamath Proof Explorer

Theorem subfacp1lem5

Description: Lemma for subfacp1 . In subfacp1lem6 we cut up the set of all derangements on 1 ... ( N + 1 ) first according to the value at 1 , and then by whether or not ( f( f1 ) ) = 1 . In this lemma, we show that the subset of all N + 1 derangements with ( f( f1 ) ) =/= 1 for fixed M = ( f1 ) is in bijection with derangements of 2 ... ( N + 1 ) , because pre-composing with the function F swaps 1 and M and turns the function into a bijection with ( f1 ) = 1 and ( fx ) =/= x for all other x , so dropping the point at 1 yields a derangement on the N remaining points. (Contributed by Mario Carneiro, 23-Jan-2015)

Ref Expression
Hypotheses derang.d ⊢ 𝐷 = ( 𝑥 ∈ Fin ↦ ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) )
subfac.n ⊢ 𝑆 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐷 ‘ ( 1 ... 𝑛 ) ) )
subfacp1lem.a ⊢ 𝐴 = { 𝑓 ∣ ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) }
subfacp1lem1.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ )
subfacp1lem1.m ⊢ ( 𝜑 → 𝑀 ∈ ( 2 ... ( 𝑁 + 1 ) ) )
subfacp1lem1.x ⊢ 𝑀 ∈ V
subfacp1lem1.k ⊢ 𝐾 = ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } )
subfacp1lem5.b ⊢ 𝐵 = { 𝑔 ∈ 𝐴 ∣ ( ( 𝑔 ‘ 1 ) = 𝑀 ∧ ( 𝑔 ‘ 𝑀 ) ≠ 1 ) }
subfacp1lem5.f ⊢ 𝐹 = ( ( I ↾ 𝐾 ) ∪ { ⟨ 1 , 𝑀 ⟩ , ⟨ 𝑀 , 1 ⟩ } )
subfacp1lem5.c ⊢ 𝐶 = { 𝑓 ∣ ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) }
Assertion subfacp1lem5 ( 𝜑 → ( ♯ ‘ 𝐵 ) = ( 𝑆 ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 derang.d ⊢ 𝐷 = ( 𝑥 ∈ Fin ↦ ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) )
2 subfac.n ⊢ 𝑆 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐷 ‘ ( 1 ... 𝑛 ) ) )
3 subfacp1lem.a ⊢ 𝐴 = { 𝑓 ∣ ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) }
4 subfacp1lem1.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ )
5 subfacp1lem1.m ⊢ ( 𝜑 → 𝑀 ∈ ( 2 ... ( 𝑁 + 1 ) ) )
6 subfacp1lem1.x ⊢ 𝑀 ∈ V
7 subfacp1lem1.k ⊢ 𝐾 = ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } )
8 subfacp1lem5.b ⊢ 𝐵 = { 𝑔 ∈ 𝐴 ∣ ( ( 𝑔 ‘ 1 ) = 𝑀 ∧ ( 𝑔 ‘ 𝑀 ) ≠ 1 ) }
9 subfacp1lem5.f ⊢ 𝐹 = ( ( I ↾ 𝐾 ) ∪ { ⟨ 1 , 𝑀 ⟩ , ⟨ 𝑀 , 1 ⟩ } )
10 subfacp1lem5.c ⊢ 𝐶 = { 𝑓 ∣ ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) }
11 fzfi ⊢ ( 1 ... ( 𝑁 + 1 ) ) ∈ Fin
12 deranglem ⊢ ( ( 1 ... ( 𝑁 + 1 ) ) ∈ Fin → { 𝑓 ∣ ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ∈ Fin )
13 11 12 ax-mp ⊢ { 𝑓 ∣ ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ∈ Fin
14 3 13 eqeltri ⊢ 𝐴 ∈ Fin
15 8 ssrab3 ⊢ 𝐵 ⊆ 𝐴
16 ssfi ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ∈ Fin )
17 14 15 16 mp2an ⊢ 𝐵 ∈ Fin
18 17 elexi ⊢ 𝐵 ∈ V
19 18 a1i ⊢ ( 𝜑 → 𝐵 ∈ V )
20 eqid ⊢ ( 𝑏 ∈ 𝐵 ↦ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ) = ( 𝑏 ∈ 𝐵 ↦ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) )
21 f1oi ⊢ ( I ↾ 𝐾 ) : 𝐾 –1-1-onto→ 𝐾
22 21 a1i ⊢ ( 𝜑 → ( I ↾ 𝐾 ) : 𝐾 –1-1-onto→ 𝐾 )
23 1 2 3 4 5 6 7 9 22 subfacp1lem2a ⊢ ( 𝜑 → ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 𝐹 ‘ 1 ) = 𝑀 ∧ ( 𝐹 ‘ 𝑀 ) = 1 ) )
24 23 simp1d ⊢ ( 𝜑 → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
25 simpr ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ 𝐵 )
26 fveq1 ⊢ ( 𝑔 = 𝑏 → ( 𝑔 ‘ 1 ) = ( 𝑏 ‘ 1 ) )
27 26 eqeq1d ⊢ ( 𝑔 = 𝑏 → ( ( 𝑔 ‘ 1 ) = 𝑀 ↔ ( 𝑏 ‘ 1 ) = 𝑀 ) )
28 fveq1 ⊢ ( 𝑔 = 𝑏 → ( 𝑔 ‘ 𝑀 ) = ( 𝑏 ‘ 𝑀 ) )
29 28 neeq1d ⊢ ( 𝑔 = 𝑏 → ( ( 𝑔 ‘ 𝑀 ) ≠ 1 ↔ ( 𝑏 ‘ 𝑀 ) ≠ 1 ) )
30 27 29 anbi12d ⊢ ( 𝑔 = 𝑏 → ( ( ( 𝑔 ‘ 1 ) = 𝑀 ∧ ( 𝑔 ‘ 𝑀 ) ≠ 1 ) ↔ ( ( 𝑏 ‘ 1 ) = 𝑀 ∧ ( 𝑏 ‘ 𝑀 ) ≠ 1 ) ) )
31 30 8 elrab2 ⊢ ( 𝑏 ∈ 𝐵 ↔ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑏 ‘ 1 ) = 𝑀 ∧ ( 𝑏 ‘ 𝑀 ) ≠ 1 ) ) )
32 25 31 sylib ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑏 ‘ 1 ) = 𝑀 ∧ ( 𝑏 ‘ 𝑀 ) ≠ 1 ) ) )
33 32 simpld ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ 𝐴 )
34 vex ⊢ 𝑏 ∈ V
35 f1oeq1 ⊢ ( 𝑓 = 𝑏 → ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ↔ 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
36 fveq1 ⊢ ( 𝑓 = 𝑏 → ( 𝑓 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) )
37 36 neeq1d ⊢ ( 𝑓 = 𝑏 → ( ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 ) )
38 37 ralbidv ⊢ ( 𝑓 = 𝑏 → ( ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 ) )
39 35 38 anbi12d ⊢ ( 𝑓 = 𝑏 → ( ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) ↔ ( 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 ) ) )
40 34 39 3 elab2 ⊢ ( 𝑏 ∈ 𝐴 ↔ ( 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 ) )
41 33 40 sylib ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 ) )
42 41 simpld ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
43 f1oco ⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
44 24 42 43 syl2an2r ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
45 f1of1 ⊢ ( ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1→ ( 1 ... ( 𝑁 + 1 ) ) )
46 df-f1 ⊢ ( ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1→ ( 1 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ∧ Fun ◡ ( 𝐹 ∘ 𝑏 ) ) )
47 46 simprbi ⊢ ( ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1→ ( 1 ... ( 𝑁 + 1 ) ) → Fun ◡ ( 𝐹 ∘ 𝑏 ) )
48 44 45 47 3syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → Fun ◡ ( 𝐹 ∘ 𝑏 ) )
49 f1ofn ⊢ ( ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) )
50 fnresdm ⊢ ( ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) = ( 𝐹 ∘ 𝑏 ) )
51 f1oeq1 ⊢ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) = ( 𝐹 ∘ 𝑏 ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ↔ ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
52 44 49 50 51 4syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ↔ ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
53 44 52 mpbird ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
54 f1ofo ⊢ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –onto→ ( 1 ... ( 𝑁 + 1 ) ) )
55 53 54 syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –onto→ ( 1 ... ( 𝑁 + 1 ) ) )
56 1ex ⊢ 1 ∈ V
57 56 56 f1osn ⊢ { ⟨ 1 , 1 ⟩ } : { 1 } –1-1-onto→ { 1 }
58 44 49 syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) )
59 4 peano2nnd ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℕ )
60 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
61 59 60 eleqtrdi ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 1 ) )
62 eluzfz1 ⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
63 61 62 syl ⊢ ( 𝜑 → 1 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
64 63 adantr ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 1 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
65 fnressn ⊢ ( ( ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) ∧ 1 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) = { ⟨ 1 , ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) ⟩ } )
66 58 64 65 syl2anc ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) = { ⟨ 1 , ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) ⟩ } )
67 f1of ⊢ ( 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
68 42 67 syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
69 68 64 fvco3d ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) = ( 𝐹 ‘ ( 𝑏 ‘ 1 ) ) )
70 32 simprd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝑏 ‘ 1 ) = 𝑀 ∧ ( 𝑏 ‘ 𝑀 ) ≠ 1 ) )
71 70 simpld ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 ‘ 1 ) = 𝑀 )
72 71 fveq2d ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑏 ‘ 1 ) ) = ( 𝐹 ‘ 𝑀 ) )
73 23 simp3d ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) = 1 )
74 73 adantr ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑀 ) = 1 )
75 69 72 74 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) = 1 )
76 75 opeq2d ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ⟨ 1 , ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) ⟩ = ⟨ 1 , 1 ⟩ )
77 76 sneqd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → { ⟨ 1 , ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) ⟩ } = { ⟨ 1 , 1 ⟩ } )
78 66 77 eqtrd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) = { ⟨ 1 , 1 ⟩ } )
79 78 f1oeq1d ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) : { 1 } –1-1-onto→ { 1 } ↔ { ⟨ 1 , 1 ⟩ } : { 1 } –1-1-onto→ { 1 } ) )
80 57 79 mpbiri ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) : { 1 } –1-1-onto→ { 1 } )
81 f1ofo ⊢ ( ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) : { 1 } –1-1-onto→ { 1 } → ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) : { 1 } –onto→ { 1 } )
82 80 81 syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) : { 1 } –onto→ { 1 } )
83 resdif ⊢ ( ( Fun ◡ ( 𝐹 ∘ 𝑏 ) ∧ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 1 ... ( 𝑁 + 1 ) ) ) : ( 1 ... ( 𝑁 + 1 ) ) –onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( ( 𝐹 ∘ 𝑏 ) ↾ { 1 } ) : { 1 } –onto→ { 1 } ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) )
84 48 55 82 83 syl3anc ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) )
85 fzsplit ⊢ ( 1 ∈ ( 1 ... ( 𝑁 + 1 ) ) → ( 1 ... ( 𝑁 + 1 ) ) = ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... ( 𝑁 + 1 ) ) ) )
86 63 85 syl ⊢ ( 𝜑 → ( 1 ... ( 𝑁 + 1 ) ) = ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... ( 𝑁 + 1 ) ) ) )
87 1z ⊢ 1 ∈ ℤ
88 fzsn ⊢ ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } )
89 87 88 ax-mp ⊢ ( 1 ... 1 ) = { 1 }
90 1p1e2 ⊢ ( 1 + 1 ) = 2
91 90 oveq1i ⊢ ( ( 1 + 1 ) ... ( 𝑁 + 1 ) ) = ( 2 ... ( 𝑁 + 1 ) )
92 89 91 uneq12i ⊢ ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... ( 𝑁 + 1 ) ) ) = ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) )
93 86 92 eqtr2di ⊢ ( 𝜑 → ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) )
94 63 snssd ⊢ ( 𝜑 → { 1 } ⊆ ( 1 ... ( 𝑁 + 1 ) ) )
95 incom ⊢ ( { 1 } ∩ ( 2 ... ( 𝑁 + 1 ) ) ) = ( ( 2 ... ( 𝑁 + 1 ) ) ∩ { 1 } )
96 1lt2 ⊢ 1 < 2
97 1re ⊢ 1 ∈ ℝ
98 2re ⊢ 2 ∈ ℝ
99 97 98 ltnlei ⊢ ( 1 < 2 ↔ ¬ 2 ≤ 1 )
100 96 99 mpbi ⊢ ¬ 2 ≤ 1
101 elfzle1 ⊢ ( 1 ∈ ( 2 ... ( 𝑁 + 1 ) ) → 2 ≤ 1 )
102 100 101 mto ⊢ ¬ 1 ∈ ( 2 ... ( 𝑁 + 1 ) )
103 disjsn ⊢ ( ( ( 2 ... ( 𝑁 + 1 ) ) ∩ { 1 } ) = ∅ ↔ ¬ 1 ∈ ( 2 ... ( 𝑁 + 1 ) ) )
104 102 103 mpbir ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∩ { 1 } ) = ∅
105 95 104 eqtri ⊢ ( { 1 } ∩ ( 2 ... ( 𝑁 + 1 ) ) ) = ∅
106 uneqdifeq ⊢ ( ( { 1 } ⊆ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( { 1 } ∩ ( 2 ... ( 𝑁 + 1 ) ) ) = ∅ ) → ( ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) ↔ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) ) )
107 94 105 106 sylancl ⊢ ( 𝜑 → ( ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) ↔ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) ) )
108 93 107 mpbid ⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) )
109 108 adantr ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) )
110 reseq2 ⊢ ( ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) )
111 110 f1oeq1d ⊢ ( ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) )
112 f1oeq2 ⊢ ( ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) )
113 f1oeq3 ⊢ ( ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) )
114 111 112 113 3bitrd ⊢ ( ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) = ( 2 ... ( 𝑁 + 1 ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) )
115 109 114 syl ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ) : ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 + 1 ) ) ∖ { 1 } ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) )
116 84 115 mpbid ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) )
117 68 adantr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
118 fzp1ss ⊢ ( 1 ∈ ℤ → ( ( 1 + 1 ) ... ( 𝑁 + 1 ) ) ⊆ ( 1 ... ( 𝑁 + 1 ) ) )
119 87 118 ax-mp ⊢ ( ( 1 + 1 ) ... ( 𝑁 + 1 ) ) ⊆ ( 1 ... ( 𝑁 + 1 ) )
120 91 119 eqsstrri ⊢ ( 2 ... ( 𝑁 + 1 ) ) ⊆ ( 1 ... ( 𝑁 + 1 ) )
121 simpr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) )
122 120 121 sselid ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
123 117 122 fvco3d ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑏 ‘ 𝑦 ) ) )
124 1 2 3 4 5 6 7 8 9 subfacp1lem4 ⊢ ( 𝜑 → ◡ 𝐹 = 𝐹 )
125 124 fveq1d ⊢ ( 𝜑 → ( ◡ 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
126 125 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
127 70 simprd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 ‘ 𝑀 ) ≠ 1 )
128 127 74 neeqtrrd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 ‘ 𝑀 ) ≠ ( 𝐹 ‘ 𝑀 ) )
129 128 adantr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑏 ‘ 𝑀 ) ≠ ( 𝐹 ‘ 𝑀 ) )
130 fveq2 ⊢ ( 𝑦 = 𝑀 → ( 𝑏 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑀 ) )
131 fveq2 ⊢ ( 𝑦 = 𝑀 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑀 ) )
132 130 131 neeq12d ⊢ ( 𝑦 = 𝑀 → ( ( 𝑏 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝑏 ‘ 𝑀 ) ≠ ( 𝐹 ‘ 𝑀 ) ) )
133 129 132 syl5ibrcom ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑦 = 𝑀 → ( 𝑏 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
134 120 sseli ⊢ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) → 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
135 41 simprd ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 )
136 135 r19.21bi ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 )
137 134 136 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 )
138 137 adantrr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝑏 ‘ 𝑦 ) ≠ 𝑦 )
139 7 eleq2i ⊢ ( 𝑦 ∈ 𝐾 ↔ 𝑦 ∈ ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } ) )
140 eldifsn ⊢ ( 𝑦 ∈ ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } ) ↔ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) )
141 139 140 bitri ⊢ ( 𝑦 ∈ 𝐾 ↔ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) )
142 1 2 3 4 5 6 7 9 22 subfacp1lem2b ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑦 ) = ( ( I ↾ 𝐾 ) ‘ 𝑦 ) )
143 fvresi ⊢ ( 𝑦 ∈ 𝐾 → ( ( I ↾ 𝐾 ) ‘ 𝑦 ) = 𝑦 )
144 143 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐾 ) → ( ( I ↾ 𝐾 ) ‘ 𝑦 ) = 𝑦 )
145 142 144 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑦 ) = 𝑦 )
146 141 145 sylan2br ⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝐹 ‘ 𝑦 ) = 𝑦 )
147 146 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝐹 ‘ 𝑦 ) = 𝑦 )
148 138 147 neeqtrrd ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝑏 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) )
149 148 expr ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑦 ≠ 𝑀 → ( 𝑏 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
150 133 149 pm2.61dne ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑏 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) )
151 150 necomd ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ 𝑦 ) ≠ ( 𝑏 ‘ 𝑦 ) )
152 126 151 eqnetrd ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑦 ) ≠ ( 𝑏 ‘ 𝑦 ) )
153 24 adantr ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
154 ffvelrn ⊢ ( ( 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝑏 ‘ 𝑦 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) )
155 68 134 154 syl2an ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑏 ‘ 𝑦 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) )
156 f1ocnvfv ⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 𝑏 ‘ 𝑦 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝑏 ‘ 𝑦 ) ) = 𝑦 → ( ◡ 𝐹 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) )
157 153 155 156 syl2an2r ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝑏 ‘ 𝑦 ) ) = 𝑦 → ( ◡ 𝐹 ‘ 𝑦 ) = ( 𝑏 ‘ 𝑦 ) ) )
158 157 necon3d ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑦 ) ≠ ( 𝑏 ‘ 𝑦 ) → ( 𝐹 ‘ ( 𝑏 ‘ 𝑦 ) ) ≠ 𝑦 ) )
159 152 158 mpd ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ ( 𝑏 ‘ 𝑦 ) ) ≠ 𝑦 )
160 123 159 eqnetrd ⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) ≠ 𝑦 )
161 160 ralrimiva ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) ≠ 𝑦 )
162 f1of ⊢ ( ( I ↾ 𝐾 ) : 𝐾 –1-1-onto→ 𝐾 → ( I ↾ 𝐾 ) : 𝐾 ⟶ 𝐾 )
163 21 162 ax-mp ⊢ ( I ↾ 𝐾 ) : 𝐾 ⟶ 𝐾
164 fzfi ⊢ ( 2 ... ( 𝑁 + 1 ) ) ∈ Fin
165 difexg ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∈ Fin → ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } ) ∈ V )
166 164 165 ax-mp ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } ) ∈ V
167 7 166 eqeltri ⊢ 𝐾 ∈ V
168 fex ⊢ ( ( ( I ↾ 𝐾 ) : 𝐾 ⟶ 𝐾 ∧ 𝐾 ∈ V ) → ( I ↾ 𝐾 ) ∈ V )
169 163 167 168 mp2an ⊢ ( I ↾ 𝐾 ) ∈ V
170 prex ⊢ { ⟨ 1 , 𝑀 ⟩ , ⟨ 𝑀 , 1 ⟩ } ∈ V
171 169 170 unex ⊢ ( ( I ↾ 𝐾 ) ∪ { ⟨ 1 , 𝑀 ⟩ , ⟨ 𝑀 , 1 ⟩ } ) ∈ V
172 9 171 eqeltri ⊢ 𝐹 ∈ V
173 172 34 coex ⊢ ( 𝐹 ∘ 𝑏 ) ∈ V
174 173 resex ⊢ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ∈ V
175 f1oeq1 ⊢ ( 𝑓 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) )
176 fveq1 ⊢ ( 𝑓 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑓 ‘ 𝑦 ) = ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) )
177 fvres ⊢ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) )
178 176 177 sylan9eq ⊢ ( ( 𝑓 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑓 ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) )
179 178 neeq1d ⊢ ( ( 𝑓 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) ≠ 𝑦 ) )
180 179 ralbidva ⊢ ( 𝑓 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) ≠ 𝑦 ) )
181 175 180 anbi12d ⊢ ( 𝑓 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) ↔ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) ≠ 𝑦 ) ) )
182 174 181 10 elab2 ⊢ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ∈ 𝐶 ↔ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) ≠ 𝑦 ) )
183 116 161 182 sylanbrc ⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ∈ 𝐶 )
184 simpr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑐 ∈ 𝐶 )
185 vex ⊢ 𝑐 ∈ V
186 f1oeq1 ⊢ ( 𝑓 = 𝑐 → ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ↔ 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) )
187 fveq1 ⊢ ( 𝑓 = 𝑐 → ( 𝑓 ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) )
188 187 neeq1d ⊢ ( 𝑓 = 𝑐 → ( ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 ) )
189 188 ralbidv ⊢ ( 𝑓 = 𝑐 → ( ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 ) )
190 186 189 anbi12d ⊢ ( 𝑓 = 𝑐 → ( ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) ↔ ( 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 ) ) )
191 185 190 10 elab2 ⊢ ( 𝑐 ∈ 𝐶 ↔ ( 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 ) )
192 184 191 sylib ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 ) )
193 192 simpld ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) )
194 f1oun ⊢ ( ( ( { ⟨ 1 , 1 ⟩ } : { 1 } –1-1-onto→ { 1 } ∧ 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) ∧ ( ( { 1 } ∩ ( 2 ... ( 𝑁 + 1 ) ) ) = ∅ ∧ ( { 1 } ∩ ( 2 ... ( 𝑁 + 1 ) ) ) = ∅ ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) )
195 105 105 194 mpanr12 ⊢ ( ( { ⟨ 1 , 1 ⟩ } : { 1 } –1-1-onto→ { 1 } ∧ 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) )
196 57 193 195 sylancr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) )
197 f1oeq2 ⊢ ( ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ↔ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ) )
198 f1oeq3 ⊢ ( ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ↔ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
199 197 198 bitrd ⊢ ( ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ↔ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
200 93 199 syl ⊢ ( 𝜑 → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ↔ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
201 200 biimpa ⊢ ( ( 𝜑 ∧ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
202 196 201 syldan ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
203 f1oco ⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
204 24 202 203 syl2an2r ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
205 f1of ⊢ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
206 202 205 syl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
207 fvco3 ⊢ ( ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) = ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
208 206 207 sylan ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) = ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
209 125 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
210 simpr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
211 93 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) )
212 210 211 eleqtrrd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → 𝑦 ∈ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) )
213 elun ⊢ ( 𝑦 ∈ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ↔ ( 𝑦 ∈ { 1 } ∨ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) )
214 212 213 sylib ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝑦 ∈ { 1 } ∨ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) )
215 nelne2 ⊢ ( ( 𝑀 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ ¬ 1 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → 𝑀 ≠ 1 )
216 5 102 215 sylancl ⊢ ( 𝜑 → 𝑀 ≠ 1 )
217 216 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑀 ≠ 1 )
218 23 simp2d ⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = 𝑀 )
219 218 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ‘ 1 ) = 𝑀 )
220 f1ofun ⊢ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) –1-1-onto→ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) → Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) )
221 196 220 syl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) )
222 ssun1 ⊢ { ⟨ 1 , 1 ⟩ } ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 )
223 56 snid ⊢ 1 ∈ { 1 }
224 56 dmsnop ⊢ dom { ⟨ 1 , 1 ⟩ } = { 1 }
225 223 224 eleqtrri ⊢ 1 ∈ dom { ⟨ 1 , 1 ⟩ }
226 funssfv ⊢ ( ( Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∧ { ⟨ 1 , 1 ⟩ } ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∧ 1 ∈ dom { ⟨ 1 , 1 ⟩ } ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) = ( { ⟨ 1 , 1 ⟩ } ‘ 1 ) )
227 222 225 226 mp3an23 ⊢ ( Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) = ( { ⟨ 1 , 1 ⟩ } ‘ 1 ) )
228 221 227 syl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) = ( { ⟨ 1 , 1 ⟩ } ‘ 1 ) )
229 56 56 fvsn ⊢ ( { ⟨ 1 , 1 ⟩ } ‘ 1 ) = 1
230 228 229 eqtrdi ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) = 1 )
231 217 219 230 3netr4d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ‘ 1 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) )
232 elsni ⊢ ( 𝑦 ∈ { 1 } → 𝑦 = 1 )
233 232 fveq2d ⊢ ( 𝑦 ∈ { 1 } → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 1 ) )
234 232 fveq2d ⊢ ( 𝑦 ∈ { 1 } → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) )
235 233 234 neeq12d ⊢ ( 𝑦 ∈ { 1 } → ( ( 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ( 𝐹 ‘ 1 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) ) )
236 231 235 syl5ibrcom ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑦 ∈ { 1 } → ( 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
237 236 imp ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ { 1 } ) → ( 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) )
238 221 adantr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) )
239 ssun2 ⊢ 𝑐 ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 )
240 239 a1i ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → 𝑐 ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) )
241 f1odm ⊢ ( 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) → dom 𝑐 = ( 2 ... ( 𝑁 + 1 ) ) )
242 193 241 syl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → dom 𝑐 = ( 2 ... ( 𝑁 + 1 ) ) )
243 242 eleq2d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑦 ∈ dom 𝑐 ↔ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) )
244 243 biimpar ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → 𝑦 ∈ dom 𝑐 )
245 funssfv ⊢ ( ( Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∧ 𝑐 ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∧ 𝑦 ∈ dom 𝑐 ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) )
246 238 240 244 245 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) )
247 f1of ⊢ ( 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) → 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) ⟶ ( 2 ... ( 𝑁 + 1 ) ) )
248 193 247 syl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) ⟶ ( 2 ... ( 𝑁 + 1 ) ) )
249 5 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑀 ∈ ( 2 ... ( 𝑁 + 1 ) ) )
250 248 249 ffvelrnd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑐 ‘ 𝑀 ) ∈ ( 2 ... ( 𝑁 + 1 ) ) )
251 nelne2 ⊢ ( ( ( 𝑐 ‘ 𝑀 ) ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ ¬ 1 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑐 ‘ 𝑀 ) ≠ 1 )
252 250 102 251 sylancl ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑐 ‘ 𝑀 ) ≠ 1 )
253 252 adantr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑐 ‘ 𝑀 ) ≠ 1 )
254 73 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ 𝑀 ) = 1 )
255 253 254 neeqtrrd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑐 ‘ 𝑀 ) ≠ ( 𝐹 ‘ 𝑀 ) )
256 fveq2 ⊢ ( 𝑦 = 𝑀 → ( 𝑐 ‘ 𝑦 ) = ( 𝑐 ‘ 𝑀 ) )
257 256 131 neeq12d ⊢ ( 𝑦 = 𝑀 → ( ( 𝑐 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝑐 ‘ 𝑀 ) ≠ ( 𝐹 ‘ 𝑀 ) ) )
258 255 257 syl5ibrcom ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑦 = 𝑀 → ( 𝑐 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
259 192 simprd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 )
260 259 r19.21bi ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 )
261 260 adantrr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 )
262 146 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝐹 ‘ 𝑦 ) = 𝑦 )
263 261 262 neeqtrrd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ ( 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑦 ≠ 𝑀 ) ) → ( 𝑐 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) )
264 263 expr ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑦 ≠ 𝑀 → ( 𝑐 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
265 258 264 pm2.61dne ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝑐 ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) )
266 246 265 eqnetrd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ≠ ( 𝐹 ‘ 𝑦 ) )
267 266 necomd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) )
268 237 267 jaodan ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ ( 𝑦 ∈ { 1 } ∨ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) )
269 214 268 syldan ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) )
270 209 269 eqnetrd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) )
271 24 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
272 206 ffvelrnda ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) )
273 f1ocnvfv ⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) = 𝑦 → ( ◡ 𝐹 ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
274 271 272 273 syl2an2r ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) = 𝑦 → ( ◡ 𝐹 ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
275 274 necon3d ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑦 ) ≠ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) → ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) ≠ 𝑦 ) )
276 270 275 mpd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) ≠ 𝑦 )
277 208 276 eqnetrd ⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) ≠ 𝑦 )
278 277 ralrimiva ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) ≠ 𝑦 )
279 snex ⊢ { ⟨ 1 , 1 ⟩ } ∈ V
280 279 185 unex ⊢ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∈ V
281 172 280 coex ⊢ ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ∈ V
282 f1oeq1 ⊢ ( 𝑓 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ↔ ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) )
283 fveq1 ⊢ ( 𝑓 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( 𝑓 ‘ 𝑦 ) = ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) )
284 283 neeq1d ⊢ ( 𝑓 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) ≠ 𝑦 ) )
285 284 ralbidv ⊢ ( 𝑓 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ↔ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) ≠ 𝑦 ) )
286 282 285 anbi12d ⊢ ( 𝑓 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) ↔ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) ≠ 𝑦 ) ) )
287 281 286 3 elab2 ⊢ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ∈ 𝐴 ↔ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑦 ) ≠ 𝑦 ) )
288 204 278 287 sylanbrc ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ∈ 𝐴 )
289 63 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 1 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
290 206 289 fvco3d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) = ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) ) )
291 230 fveq2d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) ) = ( 𝐹 ‘ 1 ) )
292 290 291 219 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) = 𝑀 )
293 120 5 sselid ⊢ ( 𝜑 → 𝑀 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
294 293 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑀 ∈ ( 1 ... ( 𝑁 + 1 ) ) )
295 206 294 fvco3d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) = ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑀 ) ) )
296 239 a1i ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑐 ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) )
297 249 242 eleqtrrd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑀 ∈ dom 𝑐 )
298 funssfv ⊢ ( ( Fun ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∧ 𝑐 ⊆ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ∧ 𝑀 ∈ dom 𝑐 ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑀 ) = ( 𝑐 ‘ 𝑀 ) )
299 221 296 297 298 syl3anc ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑀 ) = ( 𝑐 ‘ 𝑀 ) )
300 299 fveq2d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ‘ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑀 ) ) = ( 𝐹 ‘ ( 𝑐 ‘ 𝑀 ) ) )
301 295 300 eqtrd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) = ( 𝐹 ‘ ( 𝑐 ‘ 𝑀 ) ) )
302 124 fveq1d ⊢ ( 𝜑 → ( ◡ 𝐹 ‘ 1 ) = ( 𝐹 ‘ 1 ) )
303 302 218 eqtrd ⊢ ( 𝜑 → ( ◡ 𝐹 ‘ 1 ) = 𝑀 )
304 303 adantr ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ◡ 𝐹 ‘ 1 ) = 𝑀 )
305 id ⊢ ( 𝑦 = 𝑀 → 𝑦 = 𝑀 )
306 256 305 neeq12d ⊢ ( 𝑦 = 𝑀 → ( ( 𝑐 ‘ 𝑦 ) ≠ 𝑦 ↔ ( 𝑐 ‘ 𝑀 ) ≠ 𝑀 ) )
307 306 259 249 rspcdva ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑐 ‘ 𝑀 ) ≠ 𝑀 )
308 307 necomd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → 𝑀 ≠ ( 𝑐 ‘ 𝑀 ) )
309 304 308 eqnetrd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ◡ 𝐹 ‘ 1 ) ≠ ( 𝑐 ‘ 𝑀 ) )
310 120 250 sselid ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝑐 ‘ 𝑀 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) )
311 f1ocnvfv ⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 𝑐 ‘ 𝑀 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝑐 ‘ 𝑀 ) ) = 1 → ( ◡ 𝐹 ‘ 1 ) = ( 𝑐 ‘ 𝑀 ) ) )
312 24 310 311 syl2an2r ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝐹 ‘ ( 𝑐 ‘ 𝑀 ) ) = 1 → ( ◡ 𝐹 ‘ 1 ) = ( 𝑐 ‘ 𝑀 ) ) )
313 312 necon3d ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( ◡ 𝐹 ‘ 1 ) ≠ ( 𝑐 ‘ 𝑀 ) → ( 𝐹 ‘ ( 𝑐 ‘ 𝑀 ) ) ≠ 1 ) )
314 309 313 mpd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ‘ ( 𝑐 ‘ 𝑀 ) ) ≠ 1 )
315 301 314 eqnetrd ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) ≠ 1 )
316 292 315 jca ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) = 𝑀 ∧ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) ≠ 1 ) )
317 fveq1 ⊢ ( 𝑔 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( 𝑔 ‘ 1 ) = ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) )
318 317 eqeq1d ⊢ ( 𝑔 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( ( 𝑔 ‘ 1 ) = 𝑀 ↔ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) = 𝑀 ) )
319 fveq1 ⊢ ( 𝑔 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( 𝑔 ‘ 𝑀 ) = ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) )
320 319 neeq1d ⊢ ( 𝑔 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( ( 𝑔 ‘ 𝑀 ) ≠ 1 ↔ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) ≠ 1 ) )
321 318 320 anbi12d ⊢ ( 𝑔 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) → ( ( ( 𝑔 ‘ 1 ) = 𝑀 ∧ ( 𝑔 ‘ 𝑀 ) ≠ 1 ) ↔ ( ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) = 𝑀 ∧ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) ≠ 1 ) ) )
322 321 8 elrab2 ⊢ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ∈ 𝐵 ↔ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ∈ 𝐴 ∧ ( ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 1 ) = 𝑀 ∧ ( ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ‘ 𝑀 ) ≠ 1 ) ) )
323 288 316 322 sylanbrc ⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐶 ) → ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ∈ 𝐵 )
324 24 adantr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
325 f1of1 ⊢ ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1→ ( 1 ... ( 𝑁 + 1 ) ) )
326 324 325 syl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1→ ( 1 ... ( 𝑁 + 1 ) ) )
327 f1of ⊢ ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
328 324 327 syl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
329 68 adantrr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
330 328 329 fcod ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
331 206 adantrl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) )
332 cocan1 ⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 𝐹 ∘ 𝑏 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ ( 𝐹 ∘ 𝑏 ) ) = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ↔ ( 𝐹 ∘ 𝑏 ) = ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) )
333 326 330 331 332 syl3anc ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ ( 𝐹 ∘ 𝑏 ) ) = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ↔ ( 𝐹 ∘ 𝑏 ) = ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) )
334 coass ⊢ ( ( 𝐹 ∘ 𝐹 ) ∘ 𝑏 ) = ( 𝐹 ∘ ( 𝐹 ∘ 𝑏 ) )
335 124 coeq1d ⊢ ( 𝜑 → ( ◡ 𝐹 ∘ 𝐹 ) = ( 𝐹 ∘ 𝐹 ) )
336 f1ococnv1 ⊢ ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) )
337 24 336 syl ⊢ ( 𝜑 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) )
338 335 337 eqtr3d ⊢ ( 𝜑 → ( 𝐹 ∘ 𝐹 ) = ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) )
339 338 adantr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( 𝐹 ∘ 𝐹 ) = ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) )
340 339 coeq1d ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝐹 ) ∘ 𝑏 ) = ( ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) ∘ 𝑏 ) )
341 fcoi2 ⊢ ( 𝑏 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) → ( ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) ∘ 𝑏 ) = 𝑏 )
342 329 341 syl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( I ↾ ( 1 ... ( 𝑁 + 1 ) ) ) ∘ 𝑏 ) = 𝑏 )
343 340 342 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝐹 ) ∘ 𝑏 ) = 𝑏 )
344 334 343 eqtr3id ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( 𝐹 ∘ ( 𝐹 ∘ 𝑏 ) ) = 𝑏 )
345 344 eqeq1d ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ ( 𝐹 ∘ 𝑏 ) ) = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ↔ 𝑏 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ) )
346 75 adantrr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) = 1 )
347 230 adantrl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) = 1 )
348 346 347 eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) )
349 fveq2 ⊢ ( 𝑦 = 1 → ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) )
350 fveq2 ⊢ ( 𝑦 = 1 → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) )
351 349 350 eqeq12d ⊢ ( 𝑦 = 1 → ( ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) ) )
352 56 351 ralsn ⊢ ( ∀ 𝑦 ∈ { 1 } ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ( ( 𝐹 ∘ 𝑏 ) ‘ 1 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 1 ) )
353 348 352 sylibr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ∀ 𝑦 ∈ { 1 } ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) )
354 353 biantrurd ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ( ∀ 𝑦 ∈ { 1 } ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) ) )
355 ralunb ⊢ ( ∀ 𝑦 ∈ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ( ∀ 𝑦 ∈ { 1 } ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
356 354 355 bitr4di ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
357 177 adantl ⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) )
358 357 eqcomd ⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) )
359 246 adantlrl ⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) )
360 358 359 eqeq12d ⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) ∧ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) ) )
361 360 ralbidva ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) ) )
362 93 adantr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 1 ... ( 𝑁 + 1 ) ) )
363 362 raleqdv ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( { 1 } ∪ ( 2 ... ( 𝑁 + 1 ) ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
364 356 361 363 3bitr3rd ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) ) )
365 58 adantrr ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) )
366 202 adantrl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) )
367 f1ofn ⊢ ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) Fn ( 1 ... ( 𝑁 + 1 ) ) )
368 366 367 syl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) Fn ( 1 ... ( 𝑁 + 1 ) ) )
369 eqfnfv ⊢ ( ( ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) ∧ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) Fn ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ 𝑏 ) = ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ↔ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
370 365 368 369 syl2anc ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝑏 ) = ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ↔ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( ( 𝐹 ∘ 𝑏 ) ‘ 𝑦 ) = ( ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ‘ 𝑦 ) ) )
371 fnssres ⊢ ( ( ( 𝐹 ∘ 𝑏 ) Fn ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 2 ... ( 𝑁 + 1 ) ) ⊆ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) Fn ( 2 ... ( 𝑁 + 1 ) ) )
372 365 120 371 sylancl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) Fn ( 2 ... ( 𝑁 + 1 ) ) )
373 193 adantrl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) )
374 f1ofn ⊢ ( 𝑐 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) → 𝑐 Fn ( 2 ... ( 𝑁 + 1 ) ) )
375 373 374 syl ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → 𝑐 Fn ( 2 ... ( 𝑁 + 1 ) ) )
376 eqfnfv ⊢ ( ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) Fn ( 2 ... ( 𝑁 + 1 ) ) ∧ 𝑐 Fn ( 2 ... ( 𝑁 + 1 ) ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) = 𝑐 ↔ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) ) )
377 372 375 376 syl2anc ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) = 𝑐 ↔ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ‘ 𝑦 ) = ( 𝑐 ‘ 𝑦 ) ) )
378 364 370 377 3bitr4d ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝑏 ) = ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ↔ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) = 𝑐 ) )
379 eqcom ⊢ ( ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) = 𝑐 ↔ 𝑐 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) )
380 378 379 bitrdi ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( ( 𝐹 ∘ 𝑏 ) = ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ↔ 𝑐 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ) )
381 333 345 380 3bitr3d ⊢ ( ( 𝜑 ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶 ) ) → ( 𝑏 = ( 𝐹 ∘ ( { ⟨ 1 , 1 ⟩ } ∪ 𝑐 ) ) ↔ 𝑐 = ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ) )
382 20 183 323 381 f1o2d ⊢ ( 𝜑 → ( 𝑏 ∈ 𝐵 ↦ ( ( 𝐹 ∘ 𝑏 ) ↾ ( 2 ... ( 𝑁 + 1 ) ) ) ) : 𝐵 –1-1-onto→ 𝐶 )
383 19 382 hasheqf1od ⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) = ( ♯ ‘ 𝐶 ) )
384 1 2 derangen2 ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∈ Fin → ( 𝐷 ‘ ( 2 ... ( 𝑁 + 1 ) ) ) = ( 𝑆 ‘ ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) ) )
385 1 derangval ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∈ Fin → ( 𝐷 ‘ ( 2 ... ( 𝑁 + 1 ) ) ) = ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) )
386 10 fveq2i ⊢ ( ♯ ‘ 𝐶 ) = ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : ( 2 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 2 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 2 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } )
387 385 386 eqtr4di ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∈ Fin → ( 𝐷 ‘ ( 2 ... ( 𝑁 + 1 ) ) ) = ( ♯ ‘ 𝐶 ) )
388 384 387 eqtr3d ⊢ ( ( 2 ... ( 𝑁 + 1 ) ) ∈ Fin → ( 𝑆 ‘ ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) ) = ( ♯ ‘ 𝐶 ) )
389 164 388 ax-mp ⊢ ( 𝑆 ‘ ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) ) = ( ♯ ‘ 𝐶 )
390 4 60 eleqtrdi ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
391 eluzp1p1 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) )
392 390 391 syl ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) )
393 df-2 ⊢ 2 = ( 1 + 1 )
394 393 fveq2i ⊢ ( ℤ≥ ‘ 2 ) = ( ℤ≥ ‘ ( 1 + 1 ) )
395 392 394 eleqtrrdi ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 2 ) )
396 hashfz ⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 2 ) → ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) = ( ( ( 𝑁 + 1 ) − 2 ) + 1 ) )
397 395 396 syl ⊢ ( 𝜑 → ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) = ( ( ( 𝑁 + 1 ) − 2 ) + 1 ) )
398 59 nncnd ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℂ )
399 2cnd ⊢ ( 𝜑 → 2 ∈ ℂ )
400 1cnd ⊢ ( 𝜑 → 1 ∈ ℂ )
401 398 399 400 subsubd ⊢ ( 𝜑 → ( ( 𝑁 + 1 ) − ( 2 − 1 ) ) = ( ( ( 𝑁 + 1 ) − 2 ) + 1 ) )
402 2m1e1 ⊢ ( 2 − 1 ) = 1
403 402 oveq2i ⊢ ( ( 𝑁 + 1 ) − ( 2 − 1 ) ) = ( ( 𝑁 + 1 ) − 1 )
404 4 nncnd ⊢ ( 𝜑 → 𝑁 ∈ ℂ )
405 ax-1cn ⊢ 1 ∈ ℂ
406 pncan ⊢ ( ( 𝑁 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 )
407 404 405 406 sylancl ⊢ ( 𝜑 → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 )
408 403 407 syl5eq ⊢ ( 𝜑 → ( ( 𝑁 + 1 ) − ( 2 − 1 ) ) = 𝑁 )
409 397 401 408 3eqtr2d ⊢ ( 𝜑 → ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) = 𝑁 )
410 409 fveq2d ⊢ ( 𝜑 → ( 𝑆 ‘ ( ♯ ‘ ( 2 ... ( 𝑁 + 1 ) ) ) ) = ( 𝑆 ‘ 𝑁 ) )
411 389 410 eqtr3id ⊢ ( 𝜑 → ( ♯ ‘ 𝐶 ) = ( 𝑆 ‘ 𝑁 ) )
412 383 411 eqtrd ⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) = ( 𝑆 ‘ 𝑁 ) )