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Theorem elfuns 34882
Description: Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypothesis
Ref Expression
elfuns.1 𝐹 ∈ V
Assertion
Ref Expression
elfuns (𝐹 Funs ↔ Fun 𝐹)

Proof of Theorem elfuns
Dummy variables 𝑎 𝑥 𝑦 𝑧 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrel 5798 . . . . . . . . . . 11 ((Rel 𝐹𝑝𝐹) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
21ex 413 . . . . . . . . . 10 (Rel 𝐹 → (𝑝𝐹 → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩))
3 elrel 5798 . . . . . . . . . . 11 ((Rel 𝐹𝑞𝐹) → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)
43ex 413 . . . . . . . . . 10 (Rel 𝐹 → (𝑞𝐹 → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
52, 4anim12d 609 . . . . . . . . 9 (Rel 𝐹 → ((𝑝𝐹𝑞𝐹) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
65adantrd 492 . . . . . . . 8 (Rel 𝐹 → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
76pm4.71rd 563 . . . . . . 7 (Rel 𝐹 → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ((∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))))
8 19.41vvvv 1956 . . . . . . . 8 (∃𝑥𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ (∃𝑥𝑦𝑎𝑧(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
9 ee4anv 2347 . . . . . . . . 9 (∃𝑥𝑦𝑎𝑧(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ↔ (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
109anbi1i 624 . . . . . . . 8 ((∃𝑥𝑦𝑎𝑧(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ((∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
118, 10bitr2i 275 . . . . . . 7 (((∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑥𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
127, 11bitrdi 286 . . . . . 6 (Rel 𝐹 → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ∃𝑥𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))))
13122exbidv 1927 . . . . 5 (Rel 𝐹 → (∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ∃𝑝𝑞𝑥𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))))
14 excom13 2164 . . . . . 6 (∃𝑝𝑞𝑥𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑥𝑞𝑝𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
15 excom13 2164 . . . . . . . 8 (∃𝑞𝑝𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑦𝑝𝑞𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
16 exrot4 2166 . . . . . . . . . 10 (∃𝑝𝑞𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑎𝑧𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
17 excom 2162 . . . . . . . . . 10 (∃𝑎𝑧𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑧𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
18 df-3an 1089 . . . . . . . . . . . . . . . 16 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩ ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
19182exbii 1851 . . . . . . . . . . . . . . 15 (∃𝑝𝑞(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩ ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
20 opex 5464 . . . . . . . . . . . . . . . 16 𝑥, 𝑦⟩ ∈ V
21 opex 5464 . . . . . . . . . . . . . . . 16 𝑎, 𝑧⟩ ∈ V
22 eleq1 2821 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2322anbi1d 630 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝𝐹𝑞𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹)))
24 breq2 5152 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩))
2523, 24anbi12d 631 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩)))
26 eleq1 2821 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞𝐹 ↔ ⟨𝑎, 𝑧⟩ ∈ 𝐹))
2726anbi2d 629 . . . . . . . . . . . . . . . . . 18 (𝑞 = ⟨𝑎, 𝑧⟩ → ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹)))
28 breq1 5151 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩))
29 vex 3478 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
30 vex 3478 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
3121, 29, 30brtxp 34847 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦))
32 vex 3478 . . . . . . . . . . . . . . . . . . . . . . 23 𝑎 ∈ V
33 vex 3478 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
3432, 33br1steq 34737 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩1st 𝑥𝑥 = 𝑎)
35 equcom 2021 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎𝑎 = 𝑥)
3634, 35bitri 274 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩1st 𝑥𝑎 = 𝑥)
3721, 30brco 5870 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ∃𝑥(⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦))
3832, 33br2ndeq 34738 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑎, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
3938anbi1i 624 . . . . . . . . . . . . . . . . . . . . . . 23 ((⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ (𝑥 = 𝑧𝑥(V ∖ I )𝑦))
4039exbii 1850 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ ∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦))
41 breq1 5151 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦𝑧(V ∖ I )𝑦))
42 brv 5472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧V𝑦
43 brdif 5201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
4442, 43mpbiran 707 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
4530ideq 5852 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 I 𝑦𝑧 = 𝑦)
46 equcom 2021 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦𝑦 = 𝑧)
4745, 46bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 I 𝑦𝑦 = 𝑧)
4847notbii 319 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 I 𝑦 ↔ ¬ 𝑦 = 𝑧)
4944, 48bitri 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
5041, 49bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧))
5150equsexvw 2008 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦) ↔ ¬ 𝑦 = 𝑧)
5237, 40, 513bitri 296 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ¬ 𝑦 = 𝑧)
5336, 52anbi12i 627 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦) ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))
5431, 53bitri 274 . . . . . . . . . . . . . . . . . . 19 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))
5528, 54bitrdi 286 . . . . . . . . . . . . . . . . . 18 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)))
5627, 55anbi12d 631 . . . . . . . . . . . . . . . . 17 (𝑞 = ⟨𝑎, 𝑧⟩ → (((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))))
57 an12 643 . . . . . . . . . . . . . . . . 17 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)) ↔ (𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
5856, 57bitrdi 286 . . . . . . . . . . . . . . . 16 (𝑞 = ⟨𝑎, 𝑧⟩ → (((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩) ↔ (𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))))
5920, 21, 25, 58ceqsex2v 3530 . . . . . . . . . . . . . . 15 (∃𝑝𝑞(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩ ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ (𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
6019, 59bitr3i 276 . . . . . . . . . . . . . 14 (∃𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ (𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
6160exbii 1850 . . . . . . . . . . . . 13 (∃𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑎(𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
62 opeq1 4873 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → ⟨𝑎, 𝑧⟩ = ⟨𝑥, 𝑧⟩)
6362eleq1d 2818 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (⟨𝑎, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
6463anbi2d 629 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)))
6564anbi1d 630 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
6665equsexvw 2008 . . . . . . . . . . . . 13 (∃𝑎(𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
6761, 66bitri 274 . . . . . . . . . . . 12 (∃𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
6867exbii 1850 . . . . . . . . . . 11 (∃𝑧𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
69 exanali 1862 . . . . . . . . . . 11 (∃𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7068, 69bitri 274 . . . . . . . . . 10 (∃𝑧𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7116, 17, 703bitri 296 . . . . . . . . 9 (∃𝑝𝑞𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7271exbii 1850 . . . . . . . 8 (∃𝑦𝑝𝑞𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑦 ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
73 exnal 1829 . . . . . . . 8 (∃𝑦 ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7415, 72, 733bitri 296 . . . . . . 7 (∃𝑞𝑝𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7574exbii 1850 . . . . . 6 (∃𝑥𝑞𝑝𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ∃𝑥 ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
76 exnal 1829 . . . . . 6 (∃𝑥 ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7714, 75, 763bitri 296 . . . . 5 (∃𝑝𝑞𝑥𝑦𝑎𝑧((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7813, 77bitrdi 286 . . . 4 (Rel 𝐹 → (∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
7978con2bid 354 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
8079pm5.32i 575 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
81 dffun4 6559 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
82 df-funs 34828 . . . 4 Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
8382eleq2i 2825 . . 3 (𝐹 Funs 𝐹 ∈ (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))))
84 eldif 3958 . . 3 (𝐹 ∈ (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))) ↔ (𝐹 ∈ 𝒫 (V × V) ∧ ¬ 𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))))
85 elfuns.1 . . . . . 6 𝐹 ∈ V
8685elpw 4606 . . . . 5 (𝐹 ∈ 𝒫 (V × V) ↔ 𝐹 ⊆ (V × V))
87 df-rel 5683 . . . . 5 (Rel 𝐹𝐹 ⊆ (V × V))
8886, 87bitr4i 277 . . . 4 (𝐹 ∈ 𝒫 (V × V) ↔ Rel 𝐹)
8985elfix 34870 . . . . . 6 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ 𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹)
9085, 85coep 34717 . . . . . . 7 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝)
91 vex 3478 . . . . . . . . 9 𝑝 ∈ V
9285, 91coepr 34718 . . . . . . . 8 (𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9392rexbii 3094 . . . . . . 7 (∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9490, 93bitri 274 . . . . . 6 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
95 r2ex 3195 . . . . . 6 (∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝 ↔ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9689, 94, 953bitri 296 . . . . 5 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9796notbii 319 . . . 4 𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9888, 97anbi12i 627 . . 3 ((𝐹 ∈ 𝒫 (V × V) ∧ ¬ 𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))) ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
9983, 84, 983bitri 296 . 2 (𝐹 Funs ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
10080, 81, 993bitr4ri 303 1 (𝐹 Funs ↔ Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  wrex 3070  Vcvv 3474  cdif 3945  wss 3948  𝒫 cpw 4602  cop 4634   class class class wbr 5148   I cid 5573   E cep 5579   × cxp 5674  ccnv 5675  ccom 5680  Rel wrel 5681  Fun wfun 6537  1st c1st 7972  2nd c2nd 7973  ctxp 34797   Fix cfix 34802   Funs cfuns 34804
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7974  df-2nd 7975  df-txp 34821  df-fix 34826  df-funs 34828
This theorem is referenced by:  elfunsg  34883  dfrecs2  34917  dfrdg4  34918
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