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Theorem fi1uzind 13858
 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 13863) 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 dfclel 2871 . . . 4 ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
2 fi1uzind.l . . . . . . . . . . . . . 14 𝐿 ∈ ℕ0
3 nn0z 12000 . . . . . . . . . . . . . 14 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
42, 3mp1i 13 . . . . . . . . . . . . 13 (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ∈ ℤ)
5 nn0z 12000 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
65ad2antlr 726 . . . . . . . . . . . . 13 (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝑛 ∈ ℤ)
7 breq2 5035 . . . . . . . . . . . . . . . . 17 ((♯‘𝑉) = 𝑛 → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿𝑛))
87eqcoms 2806 . . . . . . . . . . . . . . . 16 (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿𝑛))
98biimpcd 252 . . . . . . . . . . . . . . 15 (𝐿 ≤ (♯‘𝑉) → (𝑛 = (♯‘𝑉) → 𝐿𝑛))
109adantr 484 . . . . . . . . . . . . . 14 ((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (♯‘𝑉) → 𝐿𝑛))
1110imp 410 . . . . . . . . . . . . 13 (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿𝑛)
12 eqeq1 2802 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐿 → (𝑥 = (♯‘𝑣) ↔ 𝐿 = (♯‘𝑣)))
1312anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (♯‘𝑣))))
1413imbi1d 345 . . . . . . . . . . . . . . 15 (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (♯‘𝑣)) → 𝜓)))
15142albidv 1924 . . . . . . . . . . . . . 14 (𝑥 = 𝐿 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (♯‘𝑣)) → 𝜓)))
16 eqeq1 2802 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝑥 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑣)))
1716anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (♯‘𝑣))))
1817imbi1d 345 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (♯‘𝑣)) → 𝜓)))
19182albidv 1924 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (♯‘𝑣)) → 𝜓)))
20 eqeq1 2802 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 + 1) → (𝑥 = (♯‘𝑣) ↔ (𝑦 + 1) = (♯‘𝑣)))
2120anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))))
2221imbi1d 345 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
23222albidv 1924 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
24 eqeq1 2802 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑛 → (𝑥 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑣)))
2524anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣))))
2625imbi1d 345 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓)))
27262albidv 1924 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓)))
28 eqcom 2805 . . . . . . . . . . . . . . . . 17 (𝐿 = (♯‘𝑣) ↔ (♯‘𝑣) = 𝐿)
29 fi1uzind.base . . . . . . . . . . . . . . . . 17 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
3028, 29sylan2b 596 . . . . . . . . . . . . . . . 16 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (♯‘𝑣)) → 𝜓)
3130gen2 1798 . . . . . . . . . . . . . . 15 𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (♯‘𝑣)) → 𝜓)
3231a1i 11 . . . . . . . . . . . . . 14 (𝐿 ∈ ℤ → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (♯‘𝑣)) → 𝜓))
33 simpl 486 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑣 = 𝑤)
34 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑒 = 𝑓)
3534sbceq1d 3725 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌[𝑓 / 𝑏]𝜌))
3633, 35sbceqbid 3727 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
37 fveq2 6650 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
3837eqeq2d 2809 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
3938adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
4036, 39anbi12d 633 . . . . . . . . . . . . . . . . 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 348 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑤𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (♯‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃)))
4342cbval2vw 2047 . . . . . . . . . . . . . . 15 (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃))
44 nn0ge0 11917 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
45 0red 10640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → 0 ∈ ℝ)
46 nn0re 11901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
472, 46mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
48 zre 11980 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
49 letr 10730 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
5045, 47, 48, 49syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
51 0nn0 11907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 ∈ ℕ0
52 pm3.22 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
53 0z 11987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ ℤ
54 eluz1 12242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ‘0))
57 eluznn0 12312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ∈ ℕ0𝑦 ∈ (ℤ‘0)) → 𝑦 ∈ ℕ0)
5851, 56, 57sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
5958ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ≤ 𝑦 → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))
6050, 59syl6com 37 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((0 ≤ 𝐿𝐿𝑦) → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0)))
6160ex 416 . . . . . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . . . . 19 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
68 eqcom 2805 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) = (♯‘𝑣) ↔ (♯‘𝑣) = (𝑦 + 1))
69 nn0p1gt0 11921 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
7069adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
71 simpr 488 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
7270, 71breqtrrd 5059 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
7368, 72sylan2b 596 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (♯‘𝑣)) → 0 < (♯‘𝑣))
7473adantrl 715 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → 0 < (♯‘𝑣))
75 hashgt0elex 13765 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ∃𝑛 𝑛𝑣)
76 fi1uzind.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
77 vex 3444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝑣 ∈ V
7877a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
79 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
80 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
81 hashdifsnp1 13857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
8268, 81syl5bi 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
8378, 79, 80, 82syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
8483imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
85 peano2nn0 11932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
8685ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
8786ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (♯‘𝑣))) → 𝑛𝑣)
9190adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
9477difexi 5197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑣 ∖ {𝑛}) ∈ V
95 fi1uzind.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝐹 ∈ V
96 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
97 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9897sbceq1d 3725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌[𝐹 / 𝑏]𝜌))
9996, 98sbceqbid 3727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌[(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
100 eqcom 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝑦 = (♯‘𝑤) ↔ (♯‘𝑤) = 𝑦)
101 fveqeq2 6659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
102100, 101syl5bb 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
103102adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
10499, 103anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦)))
105 fi1uzind.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
106104, 105imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
107106spc2gv 3549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
10894, 95, 107mp2an 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
109108expdimp 456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
110109ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
111683anbi2i 1155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
112111anbi2i 625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
113 fi1uzind.step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
114112, 113sylanb 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
11593, 110, 114syl6an 683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
116115exp41 438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
117116com15 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)))))
118117com23 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)))))
11984, 118mpcom 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))))
120119ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
121120com23 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)))))
122121ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
123122com15 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))))))
124123imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
12576, 124mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))))
126125ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛𝑣 → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))))
127126com4l 92 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛𝑣 → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))))
128127exlimiv 1931 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑛 𝑛𝑣 → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))))
12975, 128syl 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)))))
130129ex 416 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ V → (0 < (♯‘𝑣) → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))))))
131130com25 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))))))
132131elv 3446 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))))
133132imp 410 . . . . . . . . . . . . . . . . . . . . 21 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))))
134133impcom 411 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (0 < (♯‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))
13574, 134mpd 15 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))
13667, 135sylan 583 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))
137136impancom 455 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃)) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓))
138137alrimivv 1929 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓))
139138ex 416 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (♯‘𝑤)) → 𝜃) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
14043, 139syl5bi 245 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (♯‘𝑣)) → 𝜓) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
14115, 19, 23, 27, 32, 140uzind 12069 . . . . . . . . . . . . 13 ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿𝑛) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓))
1424, 6, 11, 141syl3anc 1368 . . . . . . . . . . . 12 (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓))
143 sbcex 3730 . . . . . . . . . . . . . . 15 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ V)
144 sbccom 3800 . . . . . . . . . . . . . . . 16 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌[𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
145 sbcex 3730 . . . . . . . . . . . . . . . 16 ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌𝐸 ∈ V)
146144, 145sylbi 220 . . . . . . . . . . . . . . 15 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝐸 ∈ V)
147143, 146jca 515 . . . . . . . . . . . . . 14 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
148 simpl 486 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
149 simpr 488 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑒 = 𝐸)
150149sbceq1d 3725 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌[𝐸 / 𝑏]𝜌))
151148, 150sbceqbid 3727 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
152 fveq2 6650 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
153152eqeq2d 2809 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑉 → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
154153adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
155151, 154anbi12d 633 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑉𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉))))
156 fi1uzind.1 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
157155, 156imbi12d 348 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → 𝜑)))
158157spc2gv 3549 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → 𝜑)))
159158com23 86 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑)))
160159expd 419 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (♯‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑))))
161147, 160mpcom 38 . . . . . . . . . . . . 13 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (♯‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑)))
162161imp 410 . . . . . . . . . . . 12 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑))
163142, 162syl5com 31 . . . . . . . . . . 11 (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → 𝜑))
164163exp31 423 . . . . . . . . . 10 (𝐿 ≤ (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → 𝜑))))
165164com14 96 . . . . . . . . 9 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (♯‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) → 𝜑))))
166165expcom 417 . . . . . . . 8 (𝑛 = (♯‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) → 𝜑)))))
167166com24 95 . . . . . . 7 (𝑛 = (♯‘𝑉) → (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))))
168167pm2.43i 52 . . . . . 6 (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑))))
169168imp 410 . . . . 5 ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
170169exlimiv 1931 . . . 4 (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
1711, 170sylbi 220 . . 3 ((♯‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
172 hashcl 13720 . . 3 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
173171, 172syl11 33 . 2 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
1741733imp 1108 1 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  [wsbc 3720   ∖ cdif 3878  {csn 4525   class class class wbr 5031  ‘cfv 6327  (class class class)co 7140  Fincfn 8499  ℝcr 10532  0cc0 10533  1c1 10534   + caddc 10536   < clt 10671   ≤ cle 10672  ℕ0cn0 11892  ℤcz 11976  ℤ≥cuz 12238  ♯chash 13693 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-cnex 10589  ax-resscn 10590  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-addrcl 10594  ax-mulcl 10595  ax-mulrcl 10596  ax-mulcom 10597  ax-addass 10598  ax-mulass 10599  ax-distr 10600  ax-i2m1 10601  ax-1ne0 10602  ax-1rid 10603  ax-rnegex 10604  ax-rrecex 10605  ax-cnre 10606  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609  ax-pre-mulgt0 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-dju 9321  df-card 9359  df-pnf 10673  df-mnf 10674  df-xr 10675  df-ltxr 10676  df-le 10677  df-sub 10868  df-neg 10869  df-nn 11633  df-n0 11893  df-xnn0 11963  df-z 11977  df-uz 12239  df-fz 12893  df-hash 13694 This theorem is referenced by:  brfi1uzind  13859  opfi1uzind  13862
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