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Theorem fpropnf1 7007
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 1129 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑋𝑈𝑌𝑉))
32adantr 484 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋𝑈𝑌𝑉))
4 id 22 . . . . . . . 8 (𝑍𝑊𝑍𝑊)
54, 4jca 515 . . . . . . 7 (𝑍𝑊 → (𝑍𝑊𝑍𝑊))
653ad2ant3 1132 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑍𝑊𝑍𝑊))
76adantr 484 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑍𝑊𝑍𝑊))
8 simpr 488 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝑋𝑌)
93, 7, 83jca 1125 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑋𝑈𝑌𝑉) ∧ (𝑍𝑊𝑍𝑊) ∧ 𝑋𝑌))
10 funprg 6382 . . . 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 6349 . . 3 (Fun 𝐹 ↔ Fun {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩})
1411, 13sylibr 237 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → Fun 𝐹)
15 neneq 2996 . . . 4 (𝑋𝑌 → ¬ 𝑋 = 𝑌)
1615adantl 485 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ¬ 𝑋 = 𝑌)
17 fprg 6898 . . . . . 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 2810 . . . . . 6 {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩} = 𝐹
2019feq1i 6482 . . . . 5 ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}:{𝑋, 𝑌}⟶{𝑍, 𝑍} ↔ 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
2118, 20sylib 221 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍})
22 df-f1 6333 . . . . 5 (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun 𝐹))
23 dff13 6995 . . . . . 6 (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} ↔ (𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
24 fveqeq2 6658 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑦)))
25 eqeq1 2805 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2624, 25imbi12d 348 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
2726ralbidv 3165 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦)))
28 fveqeq2 6658 . . . . . . . . . . . . 13 (𝑥 = 𝑌 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑦)))
29 eqeq1 2805 . . . . . . . . . . . . 13 (𝑥 = 𝑌 → (𝑥 = 𝑦𝑌 = 𝑦))
3028, 29imbi12d 348 . . . . . . . . . . . 12 (𝑥 = 𝑌 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)))
3130ralbidv 3165 . . . . . . . . . . 11 (𝑥 = 𝑌 → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)))
3227, 31ralprg 4595 . . . . . . . . . 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))))
33323adant3 1129 . . . . . . . . 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))))
3433adantr 484 . . . . . . . 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))))
35 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3635eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑋)))
37 eqeq2 2813 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → (𝑋 = 𝑦𝑋 = 𝑋))
3836, 37imbi12d 348 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋)))
39 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
4039eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → ((𝐹𝑋) = (𝐹𝑦) ↔ (𝐹𝑋) = (𝐹𝑌)))
41 eqeq2 2813 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
4240, 41imbi12d 348 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
4338, 42ralprg 4595 . . . . . . . . . . . 12 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ↔ (((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))
4435eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → ((𝐹𝑌) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑋)))
45 eqeq2 2813 . . . . . . . . . . . . . 14 (𝑦 = 𝑋 → (𝑌 = 𝑦𝑌 = 𝑋))
4644, 45imbi12d 348 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋)))
4739eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → ((𝐹𝑌) = (𝐹𝑦) ↔ (𝐹𝑌) = (𝐹𝑌)))
48 eqeq2 2813 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
4947, 48imbi12d 348 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌)))
5046, 49ralprg 4595 . . . . . . . . . . . 12 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦) ↔ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌))))
5143, 50anbi12d 633 . . . . . . . . . . 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)))))
52513adant3 1129 . . . . . . . . . 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)))))
5352adantr 484 . . . . . . . . 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)))))
5412fveq1i 6650 . . . . . . . . . . . . . 14 (𝐹𝑋) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋)
55 3simpb 1146 . . . . . . . . . . . . . . . . 17 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑋𝑈𝑍𝑊))
5655anim1i 617 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑋𝑈𝑍𝑊) ∧ 𝑋𝑌))
57 df-3an 1086 . . . . . . . . . . . . . . . 16 ((𝑋𝑈𝑍𝑊𝑋𝑌) ↔ ((𝑋𝑈𝑍𝑊) ∧ 𝑋𝑌))
5856, 57sylibr 237 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋𝑈𝑍𝑊𝑋𝑌))
59 fvpr1g 6935 . . . . . . . . . . . . . . 15 ((𝑋𝑈𝑍𝑊𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
6058, 59syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑋) = 𝑍)
6154, 60syl5eq 2848 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹𝑋) = 𝑍)
6212fveq1i 6650 . . . . . . . . . . . . . 14 (𝐹𝑌) = ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌)
63 3simpc 1147 . . . . . . . . . . . . . . . . 17 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (𝑌𝑉𝑍𝑊))
6463anim1i 617 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌))
65 df-3an 1086 . . . . . . . . . . . . . . . 16 ((𝑌𝑉𝑍𝑊𝑋𝑌) ↔ ((𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌))
6664, 65sylibr 237 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑌𝑉𝑍𝑊𝑋𝑌))
67 fvpr2g 6936 . . . . . . . . . . . . . . 15 ((𝑌𝑉𝑍𝑊𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
6866, 67syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ({⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}‘𝑌) = 𝑍)
6962, 68syl5req 2849 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → 𝑍 = (𝐹𝑌))
7061, 69eqtrd 2836 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹𝑋) = (𝐹𝑌))
71 idd 24 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝑋 = 𝑌𝑋 = 𝑌))
7270, 71embantd 59 . . . . . . . . . . 11 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌) → 𝑋 = 𝑌))
7372adantld 494 . . . . . . . . . 10 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) → 𝑋 = 𝑌))
7473adantrd 495 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (((((𝐹𝑋) = (𝐹𝑋) → 𝑋 = 𝑋) ∧ ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)) ∧ (((𝐹𝑌) = (𝐹𝑋) → 𝑌 = 𝑋) ∧ ((𝐹𝑌) = (𝐹𝑌) → 𝑌 = 𝑌))) → 𝑋 = 𝑌))
7553, 74sylbid 243 . . . . . . . 8 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑋) = (𝐹𝑦) → 𝑋 = 𝑦) ∧ ∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑌) = (𝐹𝑦) → 𝑌 = 𝑦)) → 𝑋 = 𝑌))
7634, 75sylbid 243 . . . . . . 7 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → 𝑋 = 𝑌))
7776adantld 494 . . . . . 6 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ {𝑋, 𝑌} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) → 𝑋 = 𝑌))
7823, 77syl5bi 245 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (𝐹:{𝑋, 𝑌}–1-1→{𝑍, 𝑍} → 𝑋 = 𝑌))
7922, 78syl5bir 246 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ((𝐹:{𝑋, 𝑌}⟶{𝑍, 𝑍} ∧ Fun 𝐹) → 𝑋 = 𝑌))
8021, 79mpand 694 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹𝑋 = 𝑌))
8116, 80mtod 201 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → ¬ Fun 𝐹)
8214, 81jca 515 1 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹 ∧ ¬ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  {cpr 4530  cop 4534  ccnv 5522  Fun wfun 6322  wf 6324  1-1wf1 6325  cfv 6328
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336
This theorem is referenced by:  ntrl2v2e  27946
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