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Theorem elfuns 35189
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 5799 . . . . . . . . . . 11 ((Rel 𝐹𝑝𝐹) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
21ex 411 . . . . . . . . . 10 (Rel 𝐹 → (𝑝𝐹 → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩))
3 elrel 5799 . . . . . . . . . . 11 ((Rel 𝐹𝑞𝐹) → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)
43ex 411 . . . . . . . . . 10 (Rel 𝐹 → (𝑞𝐹 → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
52, 4anim12d 607 . . . . . . . . 9 (Rel 𝐹 → ((𝑝𝐹𝑞𝐹) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
65adantrd 490 . . . . . . . 8 (Rel 𝐹 → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
76pm4.71rd 561 . . . . . . 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 1954 . . . . . . . 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 2345 . . . . . . . . 9 (∃𝑥𝑦𝑎𝑧(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ↔ (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
109anbi1i 622 . . . . . . . 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 1925 . . . . 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 2162 . . . . . 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 2162 . . . . . . . 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 2164 . . . . . . . . . 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 2160 . . . . . . . . . 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 1087 . . . . . . . . . . . . . . . 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 1849 . . . . . . . . . . . . . . 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 5465 . . . . . . . . . . . . . . . 16 𝑥, 𝑦⟩ ∈ V
21 opex 5465 . . . . . . . . . . . . . . . 16 𝑎, 𝑧⟩ ∈ V
22 eleq1 2819 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2322anbi1d 628 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝𝐹𝑞𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹)))
24 breq2 5153 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩))
2523, 24anbi12d 629 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩)))
26 eleq1 2819 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞𝐹 ↔ ⟨𝑎, 𝑧⟩ ∈ 𝐹))
2726anbi2d 627 . . . . . . . . . . . . . . . . . 18 (𝑞 = ⟨𝑎, 𝑧⟩ → ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹)))
28 breq1 5152 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩))
29 vex 3476 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
30 vex 3476 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
3121, 29, 30brtxp 35154 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦))
32 vex 3476 . . . . . . . . . . . . . . . . . . . . . . 23 𝑎 ∈ V
33 vex 3476 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
3432, 33br1steq 35044 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩1st 𝑥𝑥 = 𝑎)
35 equcom 2019 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎𝑎 = 𝑥)
3634, 35bitri 274 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩1st 𝑥𝑎 = 𝑥)
3721, 30brco 5871 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ∃𝑥(⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦))
3832, 33br2ndeq 35045 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑎, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
3938anbi1i 622 . . . . . . . . . . . . . . . . . . . . . . 23 ((⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ (𝑥 = 𝑧𝑥(V ∖ I )𝑦))
4039exbii 1848 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ ∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦))
41 breq1 5152 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦𝑧(V ∖ I )𝑦))
42 brv 5473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧V𝑦
43 brdif 5202 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
4442, 43mpbiran 705 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
4530ideq 5853 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 I 𝑦𝑧 = 𝑦)
46 equcom 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2006 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦) ↔ ¬ 𝑦 = 𝑧)
5237, 40, 513bitri 296 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ¬ 𝑦 = 𝑧)
5336, 52anbi12i 625 . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . 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 641 . . . . . . . . . . . . . . . . 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 3529 . . . . . . . . . . . . . . 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 1848 . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → ⟨𝑎, 𝑧⟩ = ⟨𝑥, 𝑧⟩)
6362eleq1d 2816 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (⟨𝑎, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
6463anbi2d 627 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)))
6564anbi1d 628 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
6665equsexvw 2006 . . . . . . . . . . . . 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 1848 . . . . . . . . . . 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 1860 . . . . . . . . . . 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 1848 . . . . . . . 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 1827 . . . . . . . 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 1848 . . . . . 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 1827 . . . . . 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 353 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
8079pm5.32i 573 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
81 dffun4 6560 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
82 df-funs 35135 . . . 4 Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
8382eleq2i 2823 . . 3 (𝐹 Funs 𝐹 ∈ (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))))
84 eldif 3959 . . 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 4607 . . . . 5 (𝐹 ∈ 𝒫 (V × V) ↔ 𝐹 ⊆ (V × V))
87 df-rel 5684 . . . . 5 (Rel 𝐹𝐹 ⊆ (V × V))
8886, 87bitr4i 277 . . . 4 (𝐹 ∈ 𝒫 (V × V) ↔ Rel 𝐹)
8985elfix 35177 . . . . . 6 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ 𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹)
9085, 85coep 35024 . . . . . . 7 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝)
91 vex 3476 . . . . . . . . 9 𝑝 ∈ V
9285, 91coepr 35025 . . . . . . . 8 (𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9392rexbii 3092 . . . . . . 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 3193 . . . . . 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 625 . . 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 394  w3a 1085  wal 1537   = wceq 1539  wex 1779  wcel 2104  wrex 3068  Vcvv 3472  cdif 3946  wss 3949  𝒫 cpw 4603  cop 4635   class class class wbr 5149   I cid 5574   E cep 5580   × cxp 5675  ccnv 5676  ccom 5681  Rel wrel 5682  Fun wfun 6538  1st c1st 7977  2nd c2nd 7978  ctxp 35104   Fix cfix 35109   Funs cfuns 35111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7979  df-2nd 7980  df-txp 35128  df-fix 35133  df-funs 35135
This theorem is referenced by:  elfunsg  35190  dfrecs2  35224  dfrdg4  35225
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