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Theorem fi1uzind 13077
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 13082) 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,𝑓,𝑤,𝑛,𝑣,𝑦   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 hashcl 12958 . . . 4 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
2 df-clel 2602 . . . . 5 ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
3 fi1uzind.l . . . . . . . . . . . . . . 15 𝐿 ∈ ℕ0
4 nn0z 11230 . . . . . . . . . . . . . . 15 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
53, 4mp1i 13 . . . . . . . . . . . . . 14 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ∈ ℤ)
6 nn0z 11230 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
76ad2antlr 758 . . . . . . . . . . . . . 14 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝑛 ∈ ℤ)
8 breq2 4578 . . . . . . . . . . . . . . . . . 18 ((#‘𝑉) = 𝑛 → (𝐿 ≤ (#‘𝑉) ↔ 𝐿𝑛))
98eqcoms 2614 . . . . . . . . . . . . . . . . 17 (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) ↔ 𝐿𝑛))
109biimpcd 237 . . . . . . . . . . . . . . . 16 (𝐿 ≤ (#‘𝑉) → (𝑛 = (#‘𝑉) → 𝐿𝑛))
1110adantr 479 . . . . . . . . . . . . . . 15 ((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (#‘𝑉) → 𝐿𝑛))
1211imp 443 . . . . . . . . . . . . . 14 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿𝑛)
13 eqeq1 2610 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐿 → (𝑥 = (#‘𝑣) ↔ 𝐿 = (#‘𝑣)))
1413anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣))))
1514imbi1d 329 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)))
16152albidv 1837 . . . . . . . . . . . . . . 15 (𝑥 = 𝐿 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)))
17 eqeq1 2610 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑥 = (#‘𝑣) ↔ 𝑦 = (#‘𝑣)))
1817anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣))))
1918imbi1d 329 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓)))
20192albidv 1837 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓)))
21 eqeq1 2610 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 + 1) → (𝑥 = (#‘𝑣) ↔ (𝑦 + 1) = (#‘𝑣)))
2221anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))))
2322imbi1d 329 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
24232albidv 1837 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
25 eqeq1 2610 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑛 → (𝑥 = (#‘𝑣) ↔ 𝑛 = (#‘𝑣)))
2625anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣))))
2726imbi1d 329 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓)))
28272albidv 1837 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓)))
29 eqcom 2613 . . . . . . . . . . . . . . . . . 18 (𝐿 = (#‘𝑣) ↔ (#‘𝑣) = 𝐿)
30 fi1uzind.base . . . . . . . . . . . . . . . . . 18 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
3129, 30sylan2b 490 . . . . . . . . . . . . . . . . 17 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)
3231gen2 1713 . . . . . . . . . . . . . . . 16 𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)
3332a1i 11 . . . . . . . . . . . . . . 15 (𝐿 ∈ ℤ → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓))
34 simpl 471 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑣 = 𝑤)
35 simpr 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑒 = 𝑓)
3635sbceq1d 3403 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌[𝑓 / 𝑏]𝜌))
3734, 36sbceqbid 3405 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
38 fveq2 6085 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
3938eqeq2d 2616 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
4039adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
4137, 40anbi12d 742 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) ↔ ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤))))
42 fi1uzind.2 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
4341, 42imbi12d 332 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑤𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)))
4443cbval2v 2268 . . . . . . . . . . . . . . . 16 (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) ↔ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃))
45 nn0ge0 11162 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
46 0red 9894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℤ → 0 ∈ ℝ)
47 nn0re 11145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
483, 47mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
49 zre 11211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
50 letr 9979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
5146, 48, 49, 50syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
52 0nn0 11151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ ℕ0
53 pm3.22 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
54 0z 11218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 0 ∈ ℤ
55 eluz1 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 ∈ ℤ → (𝑦 ∈ (ℤ‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
5654, 55mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
5753, 56mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ‘0))
58 eluznn0 11586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ∈ ℕ0𝑦 ∈ (ℤ‘0)) → 𝑦 ∈ ℕ0)
5952, 57, 58sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
6059ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ≤ 𝑦 → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))
6151, 60syl6com 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ≤ 𝐿𝐿𝑦) → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0)))
6261ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ≤ 𝐿 → (𝐿𝑦 → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))))
6362com14 93 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → (𝐿𝑦 → (𝑦 ∈ ℤ → (0 ≤ 𝐿𝑦 ∈ ℕ0))))
6463pm2.43a 51 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (𝐿𝑦 → (0 ≤ 𝐿𝑦 ∈ ℕ0)))
6564imp 443 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → (0 ≤ 𝐿𝑦 ∈ ℕ0))
6665com12 32 . . . . . . . . . . . . . . . . . . . . . 22 (0 ≤ 𝐿 → ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0))
673, 45, 66mp2b 10 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
68673adant1 1071 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
69 eqcom 2613 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) = (#‘𝑣) ↔ (#‘𝑣) = (𝑦 + 1))
70 nn0p1gt0 11166 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
7170adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
72 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
7371, 72breqtrrd 4602 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
7469, 73sylan2b 490 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (#‘𝑣)) → 0 < (#‘𝑣))
7574adantrl 747 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → 0 < (#‘𝑣))
76 vex 3172 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑣 ∈ V
77 hashgt0elex 12999 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛𝑣)
78 fi1uzind.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
7976a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
80 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
81 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
82 brfi1indlem 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8369, 82syl5bi 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8479, 80, 81, 83syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8584imp 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
86 peano2nn0 11177 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
8786ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
8887ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ ℕ0)
89 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((∀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)
90 simplrr 796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((∀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))
91 simprlr 798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) → 𝑛𝑣)
9291adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((∀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)
9389, 90, 923jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣))
9488, 93jca 552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)))
95 difexg 4727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
9676, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑣 ∖ {𝑛}) ∈ V
97 fi1uzind.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 𝐹 ∈ V
98 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
99 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
10099sbceq1d 3403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌[𝐹 / 𝑏]𝜌))
10198, 100sbceqbid 3405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌[(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
102 eqcom 2613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝑦 = (#‘𝑤) ↔ (#‘𝑤) = 𝑦)
103 fveq2 6085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
104103eqeq1d 2608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
105102, 104syl5bb 270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
106105adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
107101, 106anbi12d 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
108 fi1uzind.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
109107, 108imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
110109spc2gv 3265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
11196, 97, 110mp2an 703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
112111expdimp 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
113112ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∖ {𝑛})) = 𝑦𝜒))
114693anbi2i 1246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
115114anbi2i 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
116 fi1uzind.step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
117115, 116sylanb 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
11894, 113, 117syl6an 565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) = 𝑦𝜓))
119118exp41 635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
120119com15 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘(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)))))
121120com23 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘(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)))))
12285, 121mpcom 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((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))))
123122ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
124123com23 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((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)))))
125124ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (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))))))
126125com15 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ([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))))))
127126imp 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (([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)))))
12878, 127mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
129128ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
130129com4l 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
131130exlimiv 1844 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑛 𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
13277, 131syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
133132ex 448 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 ∈ V → (0 < (#‘𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
134133com25 96 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ V → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
13576, 134ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
136135imp 443 . . . . . . . . . . . . . . . . . . . . . 22 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
137136impcom 444 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))
13875, 137mpd 15 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
13968, 138sylan 486 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
140139impancom 454 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
141140alrimivv 1842 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
142141ex 448 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
14344, 142syl5bi 230 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
14416, 20, 24, 28, 33, 143uzind 11298 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿𝑛) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓))
1455, 7, 12, 144syl3anc 1317 . . . . . . . . . . . . 13 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓))
146 sbcex 3408 . . . . . . . . . . . . . . . 16 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ V)
147 sbccom 3472 . . . . . . . . . . . . . . . . 17 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌[𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
148 sbcex 3408 . . . . . . . . . . . . . . . . 17 ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌𝐸 ∈ V)
149147, 148sylbi 205 . . . . . . . . . . . . . . . 16 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝐸 ∈ V)
150146, 149jca 552 . . . . . . . . . . . . . . 15 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
151 simpl 471 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
152 simpr 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑒 = 𝐸)
153152sbceq1d 3403 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌[𝐸 / 𝑏]𝜌))
154151, 153sbceqbid 3405 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
155 fveq2 6085 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
156155eqeq2d 2616 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑉 → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
157156adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
158154, 157anbi12d 742 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉))))
159 fi1uzind.1 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
160158, 159imbi12d 332 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑉𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑)))
161160spc2gv 3265 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑)))
162161com23 83 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
163162expd 450 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))))
164150, 163mpcom 37 . . . . . . . . . . . . . 14 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
165164imp 443 . . . . . . . . . . . . 13 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))
166145, 165syl5com 31 . . . . . . . . . . . 12 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑))
167166exp31 627 . . . . . . . . . . 11 (𝐿 ≤ (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑))))
168167com14 93 . . . . . . . . . 10 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑))))
169168expcom 449 . . . . . . . . 9 (𝑛 = (#‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
170169com24 92 . . . . . . . 8 (𝑛 = (#‘𝑉) → (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
171170pm2.43i 49 . . . . . . 7 (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑))))
172171imp 443 . . . . . 6 ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
173172exlimiv 1844 . . . . 5 (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1742, 173sylbi 205 . . . 4 ((#‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1751, 174syl 17 . . 3 (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
176175com12 32 . 2 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1771763imp 1248 1 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030  wal 1472   = wceq 1474  wex 1694  wcel 1976  Vcvv 3169  [wsbc 3398  cdif 3533  {csn 4121   class class class wbr 4574  cfv 5787  (class class class)co 6524  Fincfn 7815  cr 9788  0cc0 9789  1c1 9790   + caddc 9792   < clt 9927  cle 9928  0cn0 11136  cz 11207  cuz 11516  #chash 12931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-hash 12932
This theorem is referenced by:  brfi1uzind  13078  opfi1uzind  13081
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