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Theorem elfuns 31664
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 5183 . . . . . . . . . . 11 ((Rel 𝐹𝑝𝐹) → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩)
21ex 450 . . . . . . . . . 10 (Rel 𝐹 → (𝑝𝐹 → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩))
3 elrel 5183 . . . . . . . . . . 11 ((Rel 𝐹𝑞𝐹) → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)
43ex 450 . . . . . . . . . 10 (Rel 𝐹 → (𝑞𝐹 → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
52, 4anim12d 585 . . . . . . . . 9 (Rel 𝐹 → ((𝑝𝐹𝑞𝐹) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
65adantrd 484 . . . . . . . 8 (Rel 𝐹 → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
76pm4.71rd 666 . . . . . . 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 1914 . . . . . . . 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 2183 . . . . . . . . 9 (∃𝑥𝑦𝑎𝑧(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ↔ (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
109anbi1i 730 . . . . . . . 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 265 . . . . . . 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, 11syl6bb 276 . . . . . 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 1849 . . . . 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 2041 . . . . . 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 2041 . . . . . . . 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 2043 . . . . . . . . . 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 2039 . . . . . . . . . 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 1038 . . . . . . . . . . . . . . . 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 1772 . . . . . . . . . . . . . . 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 4893 . . . . . . . . . . . . . . . 16 𝑥, 𝑦⟩ ∈ V
21 opex 4893 . . . . . . . . . . . . . . . 16 𝑎, 𝑧⟩ ∈ V
22 eleq1 2686 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2322anbi1d 740 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝𝐹𝑞𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹)))
24 breq2 4617 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩))
2523, 24anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩)))
26 eleq1 2686 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞𝐹 ↔ ⟨𝑎, 𝑧⟩ ∈ 𝐹))
2726anbi2d 739 . . . . . . . . . . . . . . . . . 18 (𝑞 = ⟨𝑎, 𝑧⟩ → ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹)))
28 breq1 4616 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩))
29 vex 3189 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
30 vex 3189 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
3121, 29, 30brtxp 31629 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦))
32 vex 3189 . . . . . . . . . . . . . . . . . . . . . . 23 𝑎 ∈ V
33 vex 3189 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
3432, 33, 29br1steq 31374 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩1st 𝑥𝑥 = 𝑎)
35 equcom 1942 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎𝑎 = 𝑥)
3634, 35bitri 264 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩1st 𝑥𝑎 = 𝑥)
3721, 30brco 5252 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ∃𝑥(⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦))
3832, 33, 29br2ndeq 31375 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑎, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
3938anbi1i 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ (𝑥 = 𝑧𝑥(V ∖ I )𝑦))
4039exbii 1771 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ ∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦))
41 breq1 4616 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦𝑧(V ∖ I )𝑦))
42 brv 31626 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧V𝑦
43 brdif 4665 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
4442, 43mpbiran 952 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
4530ideq 5234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 I 𝑦𝑧 = 𝑦)
46 equcom 1942 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦𝑦 = 𝑧)
4745, 46bitri 264 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 I 𝑦𝑦 = 𝑧)
4847notbii 310 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 I 𝑦 ↔ ¬ 𝑦 = 𝑧)
4944, 48bitri 264 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
5041, 49syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧))
5133, 50ceqsexv 3228 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦) ↔ ¬ 𝑦 = 𝑧)
5237, 40, 513bitri 286 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ¬ 𝑦 = 𝑧)
5336, 52anbi12i 732 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦) ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))
5431, 53bitri 264 . . . . . . . . . . . . . . . . . . 19 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))
5528, 54syl6bb 276 . . . . . . . . . . . . . . . . . 18 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)))
5627, 55anbi12d 746 . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . . 17 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)) ↔ (𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
5856, 57syl6bb 276 . . . . . . . . . . . . . . . 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 3231 . . . . . . . . . . . . . . 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 266 . . . . . . . . . . . . . 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 1771 . . . . . . . . . . . . 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 4370 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → ⟨𝑎, 𝑧⟩ = ⟨𝑥, 𝑧⟩)
6362eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (⟨𝑎, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
6463anbi2d 739 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)))
6564anbi1d 740 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
6629, 65ceqsexv 3228 . . . . . . . . . . . . 13 (∃𝑎(𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
6761, 66bitri 264 . . . . . . . . . . . 12 (∃𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
6867exbii 1771 . . . . . . . . . . 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 1783 . . . . . . . . . . 11 (∃𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7068, 69bitri 264 . . . . . . . . . 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 286 . . . . . . . . 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 1771 . . . . . . . 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 1751 . . . . . . . 8 (∃𝑦 ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7415, 72, 733bitri 286 . . . . . . 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 1771 . . . . . 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 1751 . . . . . 6 (∃𝑥 ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7714, 75, 763bitri 286 . . . . 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, 77syl6bb 276 . . . 4 (Rel 𝐹 → (∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
7978con2bid 344 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
8079pm5.32i 668 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
81 dffun4 5859 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
82 df-funs 31609 . . . 4 Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
8382eleq2i 2690 . . 3 (𝐹 Funs 𝐹 ∈ (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))))
84 eldif 3565 . . 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 4136 . . . . 5 (𝐹 ∈ 𝒫 (V × V) ↔ 𝐹 ⊆ (V × V))
87 df-rel 5081 . . . . 5 (Rel 𝐹𝐹 ⊆ (V × V))
8886, 87bitr4i 267 . . . 4 (𝐹 ∈ 𝒫 (V × V) ↔ Rel 𝐹)
8985elfix 31652 . . . . . 6 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ 𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹)
9085, 85coep 31349 . . . . . . 7 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝)
91 vex 3189 . . . . . . . . 9 𝑝 ∈ V
9285, 91coepr 31350 . . . . . . . 8 (𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9392rexbii 3034 . . . . . . 7 (∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9490, 93bitri 264 . . . . . 6 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
95 r2ex 3054 . . . . . 6 (∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝 ↔ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9689, 94, 953bitri 286 . . . . 5 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9796notbii 310 . . . 4 𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9888, 97anbi12i 732 . . 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 286 . 2 (𝐹 Funs ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
10080, 81, 993bitr4ri 293 1 (𝐹 Funs ↔ Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  𝒫 cpw 4130  cop 4154   class class class wbr 4613   E cep 4983   I cid 4984   × cxp 5072  ccnv 5073  ccom 5078  Rel wrel 5079  Fun wfun 5841  1st c1st 7111  2nd c2nd 7112  ctxp 31578   Fix cfix 31583   Funs cfuns 31585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-eprel 4985  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31602  df-fix 31607  df-funs 31609
This theorem is referenced by:  elfunsg  31665  dfrecs2  31699  dfrdg4  31700
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