# Metamath Proof Explorer

## Theorem fi1uzind

Description: Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with L = 0 (see opfi1ind ) or L = 1 . (Contributed by AV, 22-Oct-2020) (Revised by AV, 28-Mar-2021)

Ref Expression
Hypotheses fi1uzind.f 𝐹 ∈ V
fi1uzind.l 𝐿 ∈ ℕ0
fi1uzind.1 ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( 𝜓𝜑 ) )
fi1uzind.2 ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( 𝜓𝜃 ) )
fi1uzind.3 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛𝑣 ) → [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 )
fi1uzind.4 ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( 𝜃𝜒 ) )
fi1uzind.base ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ 𝑣 ) = 𝐿 ) → 𝜓 )
fi1uzind.step ( ( ( ( 𝑦 + 1 ) ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ∧ 𝑛𝑣 ) ) ∧ 𝜒 ) → 𝜓 )
Assertion fi1uzind ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ ( ♯ ‘ 𝑉 ) ) → 𝜑 )

### Proof

Step Hyp Ref Expression
1 fi1uzind.f 𝐹 ∈ V
2 fi1uzind.l 𝐿 ∈ ℕ0
3 fi1uzind.1 ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( 𝜓𝜑 ) )
4 fi1uzind.2 ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( 𝜓𝜃 ) )
5 fi1uzind.3 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛𝑣 ) → [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 )
6 fi1uzind.4 ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( 𝜃𝜒 ) )
7 fi1uzind.base ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ 𝑣 ) = 𝐿 ) → 𝜓 )
8 fi1uzind.step ( ( ( ( 𝑦 + 1 ) ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ∧ 𝑛𝑣 ) ) ∧ 𝜒 ) → 𝜓 )
9 dfclel ( ( ♯ ‘ 𝑉 ) ∈ ℕ0 ↔ ∃ 𝑛 ( 𝑛 = ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) )
10 nn0z ( 𝐿 ∈ ℕ0𝐿 ∈ ℤ )
11 2 10 mp1i ( ( ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝐿 ∈ ℤ )
12 nn0z ( 𝑛 ∈ ℕ0𝑛 ∈ ℤ )
13 12 ad2antlr ( ( ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝑛 ∈ ℤ )
14 breq2 ( ( ♯ ‘ 𝑉 ) = 𝑛 → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ↔ 𝐿𝑛 ) )
15 14 eqcoms ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ↔ 𝐿𝑛 ) )
16 15 biimpcd ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → ( 𝑛 = ( ♯ ‘ 𝑉 ) → 𝐿𝑛 ) )
17 16 adantr ( ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 = ( ♯ ‘ 𝑉 ) → 𝐿𝑛 ) )
18 17 imp ( ( ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝐿𝑛 )
19 eqeq1 ( 𝑥 = 𝐿 → ( 𝑥 = ( ♯ ‘ 𝑣 ) ↔ 𝐿 = ( ♯ ‘ 𝑣 ) ) )
20 19 anbi2d ( 𝑥 = 𝐿 → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) ↔ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝐿 = ( ♯ ‘ 𝑣 ) ) ) )
21 20 imbi1d ( 𝑥 = 𝐿 → ( ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝐿 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
22 21 2albidv ( 𝑥 = 𝐿 → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝐿 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
23 eqeq1 ( 𝑥 = 𝑦 → ( 𝑥 = ( ♯ ‘ 𝑣 ) ↔ 𝑦 = ( ♯ ‘ 𝑣 ) ) )
24 23 anbi2d ( 𝑥 = 𝑦 → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) ↔ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) ) )
25 24 imbi1d ( 𝑥 = 𝑦 → ( ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
26 25 2albidv ( 𝑥 = 𝑦 → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
27 eqeq1 ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 = ( ♯ ‘ 𝑣 ) ↔ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) )
28 27 anbi2d ( 𝑥 = ( 𝑦 + 1 ) → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) ↔ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) )
29 28 imbi1d ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
30 29 2albidv ( 𝑥 = ( 𝑦 + 1 ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
31 eqeq1 ( 𝑥 = 𝑛 → ( 𝑥 = ( ♯ ‘ 𝑣 ) ↔ 𝑛 = ( ♯ ‘ 𝑣 ) ) )
32 31 anbi2d ( 𝑥 = 𝑛 → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) ↔ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) ) )
33 32 imbi1d ( 𝑥 = 𝑛 → ( ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
34 33 2albidv ( 𝑥 = 𝑛 → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑥 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
35 eqcom ( 𝐿 = ( ♯ ‘ 𝑣 ) ↔ ( ♯ ‘ 𝑣 ) = 𝐿 )
36 35 7 sylan2b ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝐿 = ( ♯ ‘ 𝑣 ) ) → 𝜓 )
37 36 gen2 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝐿 = ( ♯ ‘ 𝑣 ) ) → 𝜓 )
38 37 a1i ( 𝐿 ∈ ℤ → ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝐿 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) )
39 simpl ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → 𝑣 = 𝑤 )
40 simpr ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → 𝑒 = 𝑓 )
41 40 sbceq1d ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( [ 𝑒 / 𝑏 ] 𝜌[ 𝑓 / 𝑏 ] 𝜌 ) )
42 39 41 sbceqbid ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌[ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌 ) )
43 fveq2 ( 𝑣 = 𝑤 → ( ♯ ‘ 𝑣 ) = ( ♯ ‘ 𝑤 ) )
44 43 eqeq2d ( 𝑣 = 𝑤 → ( 𝑦 = ( ♯ ‘ 𝑣 ) ↔ 𝑦 = ( ♯ ‘ 𝑤 ) ) )
45 44 adantr ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( 𝑦 = ( ♯ ‘ 𝑣 ) ↔ 𝑦 = ( ♯ ‘ 𝑤 ) ) )
46 42 45 anbi12d ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) ↔ ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) ) )
47 46 4 imbi12d ( ( 𝑣 = 𝑤𝑒 = 𝑓 ) → ( ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ) )
48 47 cbval2vw ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) )
49 nn0ge0 ( 𝐿 ∈ ℕ0 → 0 ≤ 𝐿 )
50 0red ( 𝑦 ∈ ℤ → 0 ∈ ℝ )
51 nn0re ( 𝐿 ∈ ℕ0𝐿 ∈ ℝ )
52 2 51 mp1i ( 𝑦 ∈ ℤ → 𝐿 ∈ ℝ )
53 zre ( 𝑦 ∈ ℤ → 𝑦 ∈ ℝ )
54 letr ( ( 0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 0 ≤ 𝐿𝐿𝑦 ) → 0 ≤ 𝑦 ) )
55 50 52 53 54 syl3anc ( 𝑦 ∈ ℤ → ( ( 0 ≤ 𝐿𝐿𝑦 ) → 0 ≤ 𝑦 ) )
56 0nn0 0 ∈ ℕ0
57 pm3.22 ( ( 0 ≤ 𝑦𝑦 ∈ ℤ ) → ( 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ) )
58 0z 0 ∈ ℤ
59 eluz1 ( 0 ∈ ℤ → ( 𝑦 ∈ ( ℤ ‘ 0 ) ↔ ( 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ) ) )
60 58 59 mp1i ( ( 0 ≤ 𝑦𝑦 ∈ ℤ ) → ( 𝑦 ∈ ( ℤ ‘ 0 ) ↔ ( 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ) ) )
61 57 60 mpbird ( ( 0 ≤ 𝑦𝑦 ∈ ℤ ) → 𝑦 ∈ ( ℤ ‘ 0 ) )
62 eluznn0 ( ( 0 ∈ ℕ0𝑦 ∈ ( ℤ ‘ 0 ) ) → 𝑦 ∈ ℕ0 )
63 56 61 62 sylancr ( ( 0 ≤ 𝑦𝑦 ∈ ℤ ) → 𝑦 ∈ ℕ0 )
64 63 ex ( 0 ≤ 𝑦 → ( 𝑦 ∈ ℤ → 𝑦 ∈ ℕ0 ) )
65 55 64 syl6com ( ( 0 ≤ 𝐿𝐿𝑦 ) → ( 𝑦 ∈ ℤ → ( 𝑦 ∈ ℤ → 𝑦 ∈ ℕ0 ) ) )
66 65 ex ( 0 ≤ 𝐿 → ( 𝐿𝑦 → ( 𝑦 ∈ ℤ → ( 𝑦 ∈ ℤ → 𝑦 ∈ ℕ0 ) ) ) )
67 66 com14 ( 𝑦 ∈ ℤ → ( 𝐿𝑦 → ( 𝑦 ∈ ℤ → ( 0 ≤ 𝐿𝑦 ∈ ℕ0 ) ) ) )
68 67 pm2.43a ( 𝑦 ∈ ℤ → ( 𝐿𝑦 → ( 0 ≤ 𝐿𝑦 ∈ ℕ0 ) ) )
69 68 imp ( ( 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) → ( 0 ≤ 𝐿𝑦 ∈ ℕ0 ) )
70 69 com12 ( 0 ≤ 𝐿 → ( ( 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) → 𝑦 ∈ ℕ0 ) )
71 2 49 70 mp2b ( ( 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) → 𝑦 ∈ ℕ0 )
72 71 3adant1 ( ( 𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) → 𝑦 ∈ ℕ0 )
73 eqcom ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ↔ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) )
74 nn0p1gt0 ( 𝑦 ∈ ℕ0 → 0 < ( 𝑦 + 1 ) )
75 74 adantr ( ( 𝑦 ∈ ℕ0 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ) → 0 < ( 𝑦 + 1 ) )
76 simpr ( ( 𝑦 ∈ ℕ0 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ) → ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) )
77 75 76 breqtrrd ( ( 𝑦 ∈ ℕ0 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ) → 0 < ( ♯ ‘ 𝑣 ) )
78 73 77 sylan2b ( ( 𝑦 ∈ ℕ0 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 0 < ( ♯ ‘ 𝑣 ) )
79 78 adantrl ( ( 𝑦 ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) → 0 < ( ♯ ‘ 𝑣 ) )
80 hashgt0elex ( ( 𝑣 ∈ V ∧ 0 < ( ♯ ‘ 𝑣 ) ) → ∃ 𝑛 𝑛𝑣 )
81 vex 𝑣 ∈ V
82 81 a1i ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) → 𝑣 ∈ V )
83 simpr ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) → 𝑛𝑣 )
84 simpl ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) → 𝑦 ∈ ℕ0 )
85 hashdifsnp1 ( ( 𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0 ) → ( ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) → ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) )
86 73 85 syl5bi ( ( 𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0 ) → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) )
87 82 83 84 86 syl3anc ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) )
88 87 imp ( ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 )
89 peano2nn0 ( 𝑦 ∈ ℕ0 → ( 𝑦 + 1 ) ∈ ℕ0 )
90 89 ad2antrr ( ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( 𝑦 + 1 ) ∈ ℕ0 )
91 90 ad2antlr ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → ( 𝑦 + 1 ) ∈ ℕ0 )
92 simpr ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 )
93 simplrr ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) )
94 simprlr ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) → 𝑛𝑣 )
95 94 adantr ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → 𝑛𝑣 )
96 92 93 95 3jca ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ∧ 𝑛𝑣 ) )
97 91 96 jca ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → ( ( 𝑦 + 1 ) ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ∧ 𝑛𝑣 ) ) )
98 81 difexi ( 𝑣 ∖ { 𝑛 } ) ∈ V
99 simpl ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → 𝑤 = ( 𝑣 ∖ { 𝑛 } ) )
100 simpr ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → 𝑓 = 𝐹 )
101 100 sbceq1d ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( [ 𝑓 / 𝑏 ] 𝜌[ 𝐹 / 𝑏 ] 𝜌 ) )
102 99 101 sbceqbid ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌[ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) )
103 eqcom ( 𝑦 = ( ♯ ‘ 𝑤 ) ↔ ( ♯ ‘ 𝑤 ) = 𝑦 )
104 fveqeq2 ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) → ( ( ♯ ‘ 𝑤 ) = 𝑦 ↔ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) )
105 103 104 syl5bb ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) → ( 𝑦 = ( ♯ ‘ 𝑤 ) ↔ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) )
106 105 adantr ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( 𝑦 = ( ♯ ‘ 𝑤 ) ↔ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) )
107 102 106 anbi12d ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) ↔ ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) ) )
108 107 6 imbi12d ( ( 𝑤 = ( 𝑣 ∖ { 𝑛 } ) ∧ 𝑓 = 𝐹 ) → ( ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ↔ ( ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) → 𝜒 ) ) )
109 108 spc2gv ( ( ( 𝑣 ∖ { 𝑛 } ) ∈ V ∧ 𝐹 ∈ V ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → ( ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) → 𝜒 ) ) )
110 98 1 109 mp2an ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → ( ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 ) → 𝜒 ) )
111 110 expdimp ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) → ( ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦𝜒 ) )
112 111 ad2antrr ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → ( ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦𝜒 ) )
113 73 3anbi2i ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ∧ 𝑛𝑣 ) ↔ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ∧ 𝑛𝑣 ) )
114 113 anbi2i ( ( ( 𝑦 + 1 ) ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ∧ 𝑛𝑣 ) ) ↔ ( ( 𝑦 + 1 ) ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( ♯ ‘ 𝑣 ) = ( 𝑦 + 1 ) ∧ 𝑛𝑣 ) ) )
115 114 8 sylanb ( ( ( ( 𝑦 + 1 ) ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ∧ 𝑛𝑣 ) ) ∧ 𝜒 ) → 𝜓 )
116 97 112 115 syl6an ( ( ( ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ∧ [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 ) ∧ ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) ∧ [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ) → ( ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦𝜓 ) )
117 116 exp41 ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦𝜓 ) ) ) ) )
118 117 com15 ( ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
119 118 com23 ( ( ♯ ‘ ( 𝑣 ∖ { 𝑛 } ) ) = 𝑦 → ( ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
120 88 119 mpcom ( ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) )
121 120 ex ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
122 121 com23 ( ( 𝑦 ∈ ℕ0𝑛𝑣 ) → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
123 122 ex ( 𝑦 ∈ ℕ0 → ( 𝑛𝑣 → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) ) )
124 123 com15 ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( 𝑛𝑣 → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) ) )
125 124 imp ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛𝑣 ) → ( [ ( 𝑣 ∖ { 𝑛 } ) / 𝑎 ] [ 𝐹 / 𝑏 ] 𝜌 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
126 5 125 mpd ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛𝑣 ) → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) )
127 126 ex ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( 𝑛𝑣 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
128 127 com4l ( 𝑛𝑣 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
129 128 exlimiv ( ∃ 𝑛 𝑛𝑣 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
130 80 129 syl ( ( 𝑣 ∈ V ∧ 0 < ( ♯ ‘ 𝑣 ) ) → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
131 130 ex ( 𝑣 ∈ V → ( 0 < ( ♯ ‘ 𝑣 ) → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) ) )
132 131 com25 ( 𝑣 ∈ V → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( 0 < ( ♯ ‘ 𝑣 ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) ) )
133 132 elv ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 → ( ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) → ( 𝑦 ∈ ℕ0 → ( 0 < ( ♯ ‘ 𝑣 ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) ) )
134 133 imp ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → ( 𝑦 ∈ ℕ0 → ( 0 < ( ♯ ‘ 𝑣 ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) ) )
135 134 impcom ( ( 𝑦 ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) → ( 0 < ( ♯ ‘ 𝑣 ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) ) )
136 79 135 mpd ( ( 𝑦 ∈ ℕ0 ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) )
137 72 136 sylan ( ( ( 𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) ∧ ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → 𝜓 ) )
138 137 impancom ( ( ( 𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) ∧ ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ) → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) )
139 138 alrimivv ( ( ( 𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) ∧ ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) ) → ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) )
140 139 ex ( ( 𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) → ( ∀ 𝑤𝑓 ( ( [ 𝑤 / 𝑎 ] [ 𝑓 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑤 ) ) → 𝜃 ) → ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
141 48 140 syl5bi ( ( 𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦 ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑦 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) → ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌 ∧ ( 𝑦 + 1 ) = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ) )
142 22 26 30 34 38 141 uzind ( ( 𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿𝑛 ) → ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) )
143 11 13 18 142 syl3anc ( ( ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = ( ♯ ‘ 𝑉 ) ) → ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) )
144 sbcex ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑉 ∈ V )
145 sbccom ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌[ 𝐸 / 𝑏 ] [ 𝑉 / 𝑎 ] 𝜌 )
146 sbcex ( [ 𝐸 / 𝑏 ] [ 𝑉 / 𝑎 ] 𝜌𝐸 ∈ V )
147 145 146 sylbi ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝐸 ∈ V )
148 144 147 jca ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝑉 ∈ V ∧ 𝐸 ∈ V ) )
149 simpl ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → 𝑣 = 𝑉 )
150 simpr ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → 𝑒 = 𝐸 )
151 150 sbceq1d ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( [ 𝑒 / 𝑏 ] 𝜌[ 𝐸 / 𝑏 ] 𝜌 ) )
152 149 151 sbceqbid ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌[ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 ) )
153 fveq2 ( 𝑣 = 𝑉 → ( ♯ ‘ 𝑣 ) = ( ♯ ‘ 𝑉 ) )
154 153 eqeq2d ( 𝑣 = 𝑉 → ( 𝑛 = ( ♯ ‘ 𝑣 ) ↔ 𝑛 = ( ♯ ‘ 𝑉 ) ) )
155 154 adantr ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( 𝑛 = ( ♯ ‘ 𝑣 ) ↔ 𝑛 = ( ♯ ‘ 𝑉 ) ) )
156 152 155 anbi12d ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) ↔ ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) ) )
157 156 3 imbi12d ( ( 𝑣 = 𝑉𝑒 = 𝐸 ) → ( ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) ↔ ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝜑 ) ) )
158 157 spc2gv ( ( 𝑉 ∈ V ∧ 𝐸 ∈ V ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) → ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝜑 ) ) )
159 158 com23 ( ( 𝑉 ∈ V ∧ 𝐸 ∈ V ) → ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) → 𝜑 ) ) )
160 159 expd ( ( 𝑉 ∈ V ∧ 𝐸 ∈ V ) → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) → 𝜑 ) ) ) )
161 148 160 mpcom ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) → 𝜑 ) ) )
162 161 imp ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → ( ∀ 𝑣𝑒 ( ( [ 𝑣 / 𝑎 ] [ 𝑒 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑣 ) ) → 𝜓 ) → 𝜑 ) )
163 143 162 syl5com ( ( ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = ( ♯ ‘ 𝑉 ) ) → ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝜑 ) )
164 163 exp31 ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → ( 𝑛 ∈ ℕ0 → ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → 𝜑 ) ) ) )
165 164 com14 ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑛 = ( ♯ ‘ 𝑉 ) ) → ( 𝑛 ∈ ℕ0 → ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) ) )
166 165 expcom ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝑛 ∈ ℕ0 → ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) ) ) )
167 166 com24 ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( 𝑛 ∈ ℕ0 → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) ) ) )
168 167 pm2.43i ( 𝑛 = ( ♯ ‘ 𝑉 ) → ( 𝑛 ∈ ℕ0 → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) ) )
169 168 imp ( ( 𝑛 = ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) )
170 169 exlimiv ( ∃ 𝑛 ( 𝑛 = ( ♯ ‘ 𝑉 ) ∧ 𝑛 ∈ ℕ0 ) → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) )
171 9 170 sylbi ( ( ♯ ‘ 𝑉 ) ∈ ℕ0 → ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) )
172 hashcl ( 𝑉 ∈ Fin → ( ♯ ‘ 𝑉 ) ∈ ℕ0 )
173 171 172 syl11 ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌 → ( 𝑉 ∈ Fin → ( 𝐿 ≤ ( ♯ ‘ 𝑉 ) → 𝜑 ) ) )
174 173 3imp ( ( [ 𝑉 / 𝑎 ] [ 𝐸 / 𝑏 ] 𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ ( ♯ ‘ 𝑉 ) ) → 𝜑 )