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Theorem fpropnf1 40337
 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 1074 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑋𝑈𝑌𝑉))
32adantr 480 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋𝑈𝑌𝑉))
4 id 22 . . . . . . . 8 (𝑍𝑊𝑍𝑊)
54, 4jca 553 . . . . . . 7 (𝑍𝑊 → (𝑍𝑊𝑍𝑊))
653ad2ant3 1077 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑍𝑊𝑍𝑊))
76adantr 480 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑍𝑊𝑍𝑊))
8 simpr 476 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝑋𝑌)
93, 7, 83jca 1235 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑋𝑈𝑌𝑉) ∧ (𝑍𝑊𝑍𝑊) ∧ 𝑋𝑌))
10 funprg 5854 . . . 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 5824 . . 3 (Fun 𝐹 ↔ Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
1411, 13sylibr 223 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → Fun 𝐹)
15 neneq 2788 . . . 4 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
1615adantl 481 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ¬ 𝑋 = 𝑌)
17 fprg 6327 . . . . . 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 2619 . . . . . 6 {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩} = 𝐹
2019feq1i 5949 . . . . 5 ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍} ↔ 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
2118, 20sylib 207 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
22 df-f1 5809 . . . . 5 (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun 𝐹))
23 dff13 6416 . . . . . 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 6103 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2524eqeq1d 2612 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
26 eqeq1 2614 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2725, 26imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
2827ralbidv 2969 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
29 fveq2 6103 . . . . . . . . . . . . . 14 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
3029eqeq1d 2612 . . . . . . . . . . . . 13 (𝑥 = 𝑌 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑦)))
31 eqeq1 2614 . . . . . . . . . . . . 13 (𝑥 = 𝑌 → (𝑥 = 𝑦𝑌 = 𝑦))
3230, 31imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑌 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)))
3332ralbidv 2969 . . . . . . . . . . 11 (𝑥 = 𝑌 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)))
3428, 33ralprg 4181 . . . . . . . . . 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 1074 . . . . . . . . 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 6103 . . . . . . . . . . . . . . 15 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3837eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑋)))
39 eqeq2 2621 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → (𝑋 = 𝑦𝑋 = 𝑋))
4038, 39imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋)))
41 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
4241eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
43 eqeq2 2621 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
4442, 43imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
4540, 44ralprg 4181 . . . . . . . . . . . 12 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ (((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))
4637eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → ((𝐹𝑌) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑋)))
47 eqeq2 2621 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → (𝑌 = 𝑦𝑌 = 𝑋))
4846, 47imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋)))
4941eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → ((𝐹𝑌) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑌)))
50 eqeq2 2621 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
5149, 50imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌)))
5248, 51ralprg 4181 . . . . . . . . . . . 12 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌))))
5345, 52anbi12d 743 . . . . . . . . . . 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 1074 . . . . . . . . . 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 6104 . . . . . . . . . . . . . 14 (𝐹𝑋) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋)
57 3simpb 1052 . . . . . . . . . . . . . . . . 17 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑋𝑈𝑍𝑊))
5857anim1i 590 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑋𝑈𝑍𝑊) ∧ 𝑋𝑌))
59 df-3an 1033 . . . . . . . . . . . . . . . 16 ((𝑋𝑈𝑍𝑊𝑋𝑌) ↔ ((𝑋𝑈𝑍𝑊) ∧ 𝑋𝑌))
6058, 59sylibr 223 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋𝑈𝑍𝑊𝑋𝑌))
61 fvpr1g 6363 . . . . . . . . . . . . . . 15 ((𝑋𝑈𝑍𝑊𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
6260, 61syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
6356, 62syl5eq 2656 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹𝑋) = 𝑍)
6412fveq1i 6104 . . . . . . . . . . . . . 14 (𝐹𝑌) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌)
65 3simpc 1053 . . . . . . . . . . . . . . . . 17 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑌𝑉𝑍𝑊))
6665anim1i 590 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌))
67 df-3an 1033 . . . . . . . . . . . . . . . 16 ((𝑌𝑉𝑍𝑊𝑋𝑌) ↔ ((𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌))
6866, 67sylibr 223 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑌𝑉𝑍𝑊𝑋𝑌))
69 fvpr2g 6364 . . . . . . . . . . . . . . 15 ((𝑌𝑉𝑍𝑊𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
7068, 69syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
7164, 70syl5req 2657 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝑍 = (𝐹𝑌))
7263, 71eqtrd 2644 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹𝑋) = (𝐹𝑌))
73 idd 24 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋 = 𝑌𝑋 = 𝑌))
7472, 73embantd 57 . . . . . . . . . . 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 229 . . . . . . . 8 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)) → 𝑋 = 𝑌))
7836, 77sylbid 229 . . . . . . 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 231 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} → 𝑋 = 𝑌))
8122, 80syl5bir 232 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun 𝐹) → 𝑋 = 𝑌))
8221, 81mpand 707 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹𝑋 = 𝑌))
8316, 82mtod 188 . 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {cpr 4127  ⟨cop 4131  ◡ccnv 5037  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  ntrl2v2e  41325
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