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Theorem fi1uzind 13262
 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 13267) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
Hypotheses
Ref Expression
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
Ref Expression
fi1uzind (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Distinct variable groups:   𝑎,𝑏,𝑒,𝑛,𝑣,𝑦,𝑓,𝑤   𝐸,𝑎,𝑒,𝑛,𝑣   𝐹,𝑎,𝑓,𝑤   𝑒,𝐿,𝑛,𝑣,𝑦   𝑉,𝑎,𝑏,𝑒,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣   𝜌,𝑒,𝑓,𝑛,𝑣,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑤,𝑓,𝑎,𝑏)   𝜓(𝑣,𝑒,𝑎,𝑏)   𝜒(𝑦,𝑣,𝑒,𝑛,𝑎,𝑏)   𝜃(𝑦,𝑤,𝑓,𝑎,𝑏)   𝜌(𝑎,𝑏)   𝐸(𝑦,𝑤,𝑓,𝑏)   𝐹(𝑦,𝑣,𝑒,𝑛,𝑏)   𝐿(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem fi1uzind
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-clel 2616 . . . 4 ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
2 fi1uzind.l . . . . . . . . . . . . . 14 𝐿 ∈ ℕ0
3 nn0z 11385 . . . . . . . . . . . . . 14 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
42, 3mp1i 13 . . . . . . . . . . . . 13 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ∈ ℤ)
5 nn0z 11385 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
65ad2antlr 762 . . . . . . . . . . . . 13 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝑛 ∈ ℤ)
7 breq2 4648 . . . . . . . . . . . . . . . . 17 ((#‘𝑉) = 𝑛 → (𝐿 ≤ (#‘𝑉) ↔ 𝐿𝑛))
87eqcoms 2628 . . . . . . . . . . . . . . . 16 (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) ↔ 𝐿𝑛))
98biimpcd 239 . . . . . . . . . . . . . . 15 (𝐿 ≤ (#‘𝑉) → (𝑛 = (#‘𝑉) → 𝐿𝑛))
109adantr 481 . . . . . . . . . . . . . 14 ((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (#‘𝑉) → 𝐿𝑛))
1110imp 445 . . . . . . . . . . . . 13 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿𝑛)
12 eqeq1 2624 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐿 → (𝑥 = (#‘𝑣) ↔ 𝐿 = (#‘𝑣)))
1312anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣))))
1413imbi1d 331 . . . . . . . . . . . . . . 15 (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)))
15142albidv 1849 . . . . . . . . . . . . . 14 (𝑥 = 𝐿 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)))
16 eqeq1 2624 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝑥 = (#‘𝑣) ↔ 𝑦 = (#‘𝑣)))
1716anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣))))
1817imbi1d 331 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓)))
19182albidv 1849 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓)))
20 eqeq1 2624 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 + 1) → (𝑥 = (#‘𝑣) ↔ (𝑦 + 1) = (#‘𝑣)))
2120anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))))
2221imbi1d 331 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
23222albidv 1849 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
24 eqeq1 2624 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑛 → (𝑥 = (#‘𝑣) ↔ 𝑛 = (#‘𝑣)))
2524anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣))))
2625imbi1d 331 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓)))
27262albidv 1849 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓)))
28 eqcom 2627 . . . . . . . . . . . . . . . . 17 (𝐿 = (#‘𝑣) ↔ (#‘𝑣) = 𝐿)
29 fi1uzind.base . . . . . . . . . . . . . . . . 17 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
3028, 29sylan2b 492 . . . . . . . . . . . . . . . 16 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)
3130gen2 1721 . . . . . . . . . . . . . . 15 𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)
3231a1i 11 . . . . . . . . . . . . . 14 (𝐿 ∈ ℤ → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓))
33 simpl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑣 = 𝑤)
34 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑒 = 𝑓)
3534sbceq1d 3434 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌[𝑓 / 𝑏]𝜌))
3633, 35sbceqbid 3436 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
37 fveq2 6178 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
3837eqeq2d 2630 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
3938adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
4036, 39anbi12d 746 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑤𝑒 = 𝑓) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) ↔ ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤))))
41 fi1uzind.2 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
4240, 41imbi12d 334 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑤𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)))
4342cbval2v 2283 . . . . . . . . . . . . . . 15 (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) ↔ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃))
44 nn0ge0 11303 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
45 0red 10026 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → 0 ∈ ℝ)
46 nn0re 11286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
472, 46mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
48 zre 11366 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
49 letr 10116 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
5045, 47, 48, 49syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
51 0nn0 11292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 ∈ ℕ0
52 pm3.22 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
53 0z 11373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ ℤ
54 eluz1 11676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 ∈ ℤ → (𝑦 ∈ (ℤ‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
5553, 54mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
5652, 55mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ‘0))
57 eluznn0 11742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ∈ ℕ0𝑦 ∈ (ℤ‘0)) → 𝑦 ∈ ℕ0)
5851, 56, 57sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
5958ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ≤ 𝑦 → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))
6050, 59syl6com 37 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((0 ≤ 𝐿𝐿𝑦) → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0)))
6160ex 450 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ≤ 𝐿 → (𝐿𝑦 → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))))
6261com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (𝐿𝑦 → (𝑦 ∈ ℤ → (0 ≤ 𝐿𝑦 ∈ ℕ0))))
6362pm2.43a 54 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℤ → (𝐿𝑦 → (0 ≤ 𝐿𝑦 ∈ ℕ0)))
6463imp 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → (0 ≤ 𝐿𝑦 ∈ ℕ0))
6564com12 32 . . . . . . . . . . . . . . . . . . . . 21 (0 ≤ 𝐿 → ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0))
662, 44, 65mp2b 10 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
67663adant1 1077 . . . . . . . . . . . . . . . . . . 19 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
68 eqcom 2627 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) = (#‘𝑣) ↔ (#‘𝑣) = (𝑦 + 1))
69 nn0p1gt0 11307 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
7069adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
71 simpr 477 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
7270, 71breqtrrd 4672 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
7368, 72sylan2b 492 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (#‘𝑣)) → 0 < (#‘𝑣))
7473adantrl 751 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → 0 < (#‘𝑣))
75 vex 3198 . . . . . . . . . . . . . . . . . . . . . . 23 𝑣 ∈ V
76 hashgt0elex 13172 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛𝑣)
77 fi1uzind.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
7875a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
79 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
80 simpl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
81 brfi1indlem 13261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8268, 81syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8378, 79, 80, 82syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8483imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
85 peano2nn0 11318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
8685ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
8786ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((∀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)
88 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((∀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)
89 simplrr 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((∀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))
90 simprlr 802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) → 𝑛𝑣)
9190adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((∀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)
9288, 89, 913jca 1240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((∀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))
9387, 92jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((∀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)))
94 difexg 4799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
9575, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑣 ∖ {𝑛}) ∈ V
96 fi1uzind.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝐹 ∈ V
97 simpl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
98 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9998sbceq1d 3434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌[𝐹 / 𝑏]𝜌))
10097, 99sbceqbid 3436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌[(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
101 eqcom 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝑦 = (#‘𝑤) ↔ (#‘𝑤) = 𝑦)
102 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
103102eqeq1d 2622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
104101, 103syl5bb 272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
105104adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
106100, 105anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
107 fi1uzind.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
108106, 107imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
109108spc2gv 3291 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
11095, 96, 109mp2an 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
111110expdimp 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
112111ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((∀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))
113683anbi2i 1252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
114113anbi2i 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((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 fi1uzind.step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
116114, 115sylanb 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
11793, 112, 116syl6an 567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((∀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))
118117exp41 637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (∀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)))))
119118com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘(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)))))
120119com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘(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)))))
12184, 120mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((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))))
122121ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((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)))))
123122com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((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)))))
124123ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (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))))))
125124com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ([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))))))
126125imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (([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)))))
12777, 126mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
128127ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
129128com4l 92 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
130129exlimiv 1856 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑛 𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
13176, 130syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
132131ex 450 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ V → (0 < (#‘𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
133132com25 99 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ V → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
13475, 133ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
135134imp 445 . . . . . . . . . . . . . . . . . . . . 21 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
136135impcom 446 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))
13774, 136mpd 15 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
13867, 137sylan 488 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
139138impancom 456 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
140139alrimivv 1854 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
141140ex 450 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
14243, 141syl5bi 232 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
14315, 19, 23, 27, 32, 142uzind 11454 . . . . . . . . . . . . 13 ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿𝑛) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓))
1444, 6, 11, 143syl3anc 1324 . . . . . . . . . . . 12 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓))
145 sbcex 3439 . . . . . . . . . . . . . . 15 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ V)
146 sbccom 3503 . . . . . . . . . . . . . . . 16 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌[𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
147 sbcex 3439 . . . . . . . . . . . . . . . 16 ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌𝐸 ∈ V)
148146, 147sylbi 207 . . . . . . . . . . . . . . 15 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝐸 ∈ V)
149145, 148jca 554 . . . . . . . . . . . . . 14 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
150 simpl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
151 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑒 = 𝐸)
152151sbceq1d 3434 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌[𝐸 / 𝑏]𝜌))
153150, 152sbceqbid 3436 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
154 fveq2 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
155154eqeq2d 2630 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑉 → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
156155adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
157153, 156anbi12d 746 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑉𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉))))
158 fi1uzind.1 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
159157, 158imbi12d 334 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑)))
160159spc2gv 3291 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑)))
161160com23 86 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
162161expd 452 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))))
163149, 162mpcom 38 . . . . . . . . . . . . 13 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
164163imp 445 . . . . . . . . . . . 12 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))
165144, 164syl5com 31 . . . . . . . . . . 11 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑))
166165exp31 629 . . . . . . . . . 10 (𝐿 ≤ (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑))))
167166com14 96 . . . . . . . . 9 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑))))
168167expcom 451 . . . . . . . 8 (𝑛 = (#‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
169168com24 95 . . . . . . 7 (𝑛 = (#‘𝑉) → (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
170169pm2.43i 52 . . . . . 6 (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑))))
171170imp 445 . . . . 5 ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
172171exlimiv 1856 . . . 4 (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1731, 172sylbi 207 . . 3 ((#‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
174 hashcl 13130 . . 3 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
175173, 174syl11 33 . 2 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1761753imp 1254 1 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036  ∀wal 1479   = wceq 1481  ∃wex 1702   ∈ wcel 1988  Vcvv 3195  [wsbc 3429   ∖ cdif 3564  {csn 4168   class class class wbr 4644  ‘cfv 5876  (class class class)co 6635  Fincfn 7940  ℝcr 9920  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059   ≤ cle 10060  ℕ0cn0 11277  ℤcz 11362  ℤ≥cuz 11672  #chash 13100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101 This theorem is referenced by:  brfi1uzind  13263  opfi1uzind  13266
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