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Theorem fpropnf1 6564
Description: A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
Hypothesis
Ref Expression
fpropnf1.f 𝐹 = {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}
Assertion
Ref Expression
fpropnf1 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹 ∧ ¬ Fun 𝐹))

Proof of Theorem fpropnf1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . 7 ((𝑋𝑈𝑌𝑉) → (𝑋𝑈𝑌𝑉))
213adant3 1101 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑋𝑈𝑌𝑉))
32adantr 480 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋𝑈𝑌𝑉))
4 id 22 . . . . . . . 8 (𝑍𝑊𝑍𝑊)
54, 4jca 553 . . . . . . 7 (𝑍𝑊 → (𝑍𝑊𝑍𝑊))
653ad2ant3 1104 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑍𝑊𝑍𝑊))
76adantr 480 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑍𝑊𝑍𝑊))
8 simpr 476 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝑋𝑌)
93, 7, 83jca 1261 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑋𝑈𝑌𝑉) ∧ (𝑍𝑊𝑍𝑊) ∧ 𝑋𝑌))
10 funprg 5978 . . . 4 (((𝑋𝑈𝑌𝑉) ∧ (𝑍𝑊𝑍𝑊) ∧ 𝑋𝑌) → Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
119, 10syl 17 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
12 fpropnf1.f . . . 4 𝐹 = {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}
1312funeqi 5947 . . 3 (Fun 𝐹 ↔ Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
1411, 13sylibr 224 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → Fun 𝐹)
15 neneq 2829 . . . 4 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
1615adantl 481 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ¬ 𝑋 = 𝑌)
17 fprg 6462 . . . . . 6 (((𝑋𝑈𝑌𝑉) ∧ (𝑍𝑊𝑍𝑊) ∧ 𝑋𝑌) → {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍})
189, 17syl 17 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍})
1912eqcomi 2660 . . . . . 6 {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩} = 𝐹
2019feq1i 6074 . . . . 5 ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍} ↔ 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
2118, 20sylib 208 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
22 df-f1 5931 . . . . 5 (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun 𝐹))
23 dff13 6552 . . . . . 6 (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
24 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2524eqeq1d 2653 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
26 eqeq1 2655 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2725, 26imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
2827ralbidv 3015 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
29 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
3029eqeq1d 2653 . . . . . . . . . . . . 13 (𝑥 = 𝑌 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑦)))
31 eqeq1 2655 . . . . . . . . . . . . 13 (𝑥 = 𝑌 → (𝑥 = 𝑦𝑌 = 𝑦))
3230, 31imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑌 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)))
3332ralbidv 3015 . . . . . . . . . . 11 (𝑥 = 𝑌 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)))
3428, 33ralprg 4266 . . . . . . . . . 10 ((𝑋𝑈𝑌𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦))))
35343adant3 1101 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦))))
3635adantr 480 . . . . . . . 8 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦))))
37 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3837eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑋)))
39 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → (𝑋 = 𝑦𝑋 = 𝑋))
4038, 39imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋)))
41 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
4241eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
43 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
4442, 43imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
4540, 44ralprg 4266 . . . . . . . . . . . 12 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ (((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))
4637eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → ((𝐹𝑌) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑋)))
47 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → (𝑌 = 𝑦𝑌 = 𝑋))
4846, 47imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋)))
4941eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → ((𝐹𝑌) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑌)))
50 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
5149, 50imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌)))
5248, 51ralprg 4266 . . . . . . . . . . . 12 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌))))
5345, 52anbi12d 747 . . . . . . . . . . 11 ((𝑋𝑈𝑌𝑉) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)) ↔ ((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌)))))
54533adant3 1101 . . . . . . . . . 10 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)) ↔ ((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌)))))
5554adantr 480 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)) ↔ ((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌)))))
5612fveq1i 6230 . . . . . . . . . . . . . 14 (𝐹𝑋) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋)
57 3simpb 1079 . . . . . . . . . . . . . . . . 17 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑋𝑈𝑍𝑊))
5857anim1i 591 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑋𝑈𝑍𝑊) ∧ 𝑋𝑌))
59 df-3an 1056 . . . . . . . . . . . . . . . 16 ((𝑋𝑈𝑍𝑊𝑋𝑌) ↔ ((𝑋𝑈𝑍𝑊) ∧ 𝑋𝑌))
6058, 59sylibr 224 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋𝑈𝑍𝑊𝑋𝑌))
61 fvpr1g 6499 . . . . . . . . . . . . . . 15 ((𝑋𝑈𝑍𝑊𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
6260, 61syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
6356, 62syl5eq 2697 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹𝑋) = 𝑍)
6412fveq1i 6230 . . . . . . . . . . . . . 14 (𝐹𝑌) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌)
65 3simpc 1080 . . . . . . . . . . . . . . . . 17 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑌𝑉𝑍𝑊))
6665anim1i 591 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌))
67 df-3an 1056 . . . . . . . . . . . . . . . 16 ((𝑌𝑉𝑍𝑊𝑋𝑌) ↔ ((𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌))
6866, 67sylibr 224 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑌𝑉𝑍𝑊𝑋𝑌))
69 fvpr2g 6500 . . . . . . . . . . . . . . 15 ((𝑌𝑉𝑍𝑊𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
7068, 69syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
7164, 70syl5req 2698 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝑍 = (𝐹𝑌))
7263, 71eqtrd 2685 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹𝑋) = (𝐹𝑌))
73 idd 24 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋 = 𝑌𝑋 = 𝑌))
7472, 73embantd 59 . . . . . . . . . . 11 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌) → 𝑋 = 𝑌))
7574adantld 482 . . . . . . . . . 10 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) → 𝑋 = 𝑌))
7675adantrd 483 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌))) → 𝑋 = 𝑌))
7755, 76sylbid 230 . . . . . . . 8 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)) → 𝑋 = 𝑌))
7836, 77sylbid 230 . . . . . . 7 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → 𝑋 = 𝑌))
7978adantld 482 . . . . . 6 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) → 𝑋 = 𝑌))
8023, 79syl5bi 232 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} → 𝑋 = 𝑌))
8122, 80syl5bir 233 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun 𝐹) → 𝑋 = 𝑌))
8221, 81mpand 711 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹𝑋 = 𝑌))
8316, 82mtod 189 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ¬ Fun 𝐹)
8414, 83jca 553 1 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹 ∧ ¬ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  {cpr 4212  cop 4216  ccnv 5142  Fun wfun 5920  wf 5922  1-1wf1 5923  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by:  ntrl2v2e  27136
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