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Theorem elfuns 34887
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 414 . . . . . . . . . 10 (Rel 𝐹 → (𝑝𝐹 → ∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩))
3 elrel 5799 . . . . . . . . . . 11 ((Rel 𝐹𝑞𝐹) → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)
43ex 414 . . . . . . . . . 10 (Rel 𝐹 → (𝑞𝐹 → ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
52, 4anim12d 610 . . . . . . . . 9 (Rel 𝐹 → ((𝑝𝐹𝑞𝐹) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
65adantrd 493 . . . . . . . 8 (Rel 𝐹 → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) → (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩)))
76pm4.71rd 564 . . . . . . 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 1957 . . . . . . . 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 2348 . . . . . . . . 9 (∃𝑥𝑦𝑎𝑧(𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ↔ (∃𝑥𝑦 𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑧 𝑞 = ⟨𝑎, 𝑧⟩))
109anbi1i 625 . . . . . . . 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 276 . . . . . . 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 287 . . . . . 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 1928 . . . . 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 2165 . . . . . 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 2165 . . . . . . . 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 2167 . . . . . . . . . 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 2163 . . . . . . . . . 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 1090 . . . . . . . . . . . . . . . 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 1852 . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2322anbi1d 631 . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩)))
26 eleq1 2822 . . . . . . . . . . . . . . . . . . 19 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞𝐹 ↔ ⟨𝑎, 𝑧⟩ ∈ 𝐹))
2726anbi2d 630 . . . . . . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
30 vex 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
3121, 29, 30brtxp 34852 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦))
32 vex 3479 . . . . . . . . . . . . . . . . . . . . . . 23 𝑎 ∈ V
33 vex 3479 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
3432, 33br1steq 34742 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑎, 𝑧⟩1st 𝑥𝑥 = 𝑎)
35 equcom 2022 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎𝑎 = 𝑥)
3634, 35bitri 275 . . . . . . . . . . . . . . . . . . . . 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 34743 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑎, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
3938anbi1i 625 . . . . . . . . . . . . . . . . . . . . . . 23 ((⟨𝑎, 𝑧⟩2nd 𝑥𝑥(V ∖ I )𝑦) ↔ (𝑥 = 𝑧𝑥(V ∖ I )𝑦))
4039exbii 1851 . . . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
4530ideq 5853 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 I 𝑦𝑧 = 𝑦)
46 equcom 2022 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦𝑦 = 𝑧)
4745, 46bitri 275 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 I 𝑦𝑦 = 𝑧)
4847notbii 320 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 I 𝑦 ↔ ¬ 𝑦 = 𝑧)
4944, 48bitri 275 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
5041, 49bitrdi 287 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧))
5150equsexvw 2009 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥(𝑥 = 𝑧𝑥(V ∖ I )𝑦) ↔ ¬ 𝑦 = 𝑧)
5237, 40, 513bitri 297 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦 ↔ ¬ 𝑦 = 𝑧)
5336, 52anbi12i 628 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑎, 𝑧⟩1st 𝑥 ∧ ⟨𝑎, 𝑧⟩((V ∖ I ) ∘ 2nd )𝑦) ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))
5431, 53bitri 275 . . . . . . . . . . . . . . . . . . 19 (⟨𝑎, 𝑧⟩(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))
5528, 54bitrdi 287 . . . . . . . . . . . . . . . . . 18 (𝑞 = ⟨𝑎, 𝑧⟩ → (𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)))
5627, 55anbi12d 632 . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . 17 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)) ↔ (𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
5856, 57bitrdi 287 . . . . . . . . . . . . . . . 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 3531 . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . 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 1851 . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (⟨𝑎, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
6463anbi2d 630 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)))
6564anbi1d 631 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))
6665equsexvw 2009 . . . . . . . . . . . . 13 (∃𝑎(𝑎 = 𝑥 ∧ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑎, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
6761, 66bitri 275 . . . . . . . . . . . 12 (∃𝑎𝑝𝑞((𝑝 = ⟨𝑥, 𝑦⟩ ∧ 𝑞 = ⟨𝑎, 𝑧⟩) ∧ ((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))
6867exbii 1851 . . . . . . . . . . 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 1863 . . . . . . . . . . 11 (∃𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7068, 69bitri 275 . . . . . . . . . 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 297 . . . . . . . . 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 1851 . . . . . . . 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 1830 . . . . . . . 8 (∃𝑦 ¬ ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7415, 72, 733bitri 297 . . . . . . 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 1851 . . . . . 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 1830 . . . . . 6 (∃𝑥 ¬ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
7714, 75, 763bitri 297 . . . . 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 287 . . . 4 (Rel 𝐹 → (∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝) ↔ ¬ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
7978con2bid 355 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
8079pm5.32i 576 . 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 34833 . . . 4 Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
8382eleq2i 2826 . . 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 278 . . . 4 (𝐹 ∈ 𝒫 (V × V) ↔ Rel 𝐹)
8985elfix 34875 . . . . . 6 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ 𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹)
9085, 85coep 34722 . . . . . . 7 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝)
91 vex 3479 . . . . . . . . 9 𝑝 ∈ V
9285, 91coepr 34723 . . . . . . . 8 (𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9392rexbii 3095 . . . . . . 7 (∃𝑝𝐹 𝐹((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )𝑝 ↔ ∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
9490, 93bitri 275 . . . . . 6 (𝐹( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E ))𝐹 ↔ ∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)
95 r2ex 3196 . . . . . 6 (∃𝑝𝐹𝑞𝐹 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝 ↔ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9689, 94, 953bitri 297 . . . . 5 (𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9796notbii 320 . . . 4 𝐹 Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )) ↔ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝))
9888, 97anbi12i 628 . . 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 297 . 2 (𝐹 Funs ↔ (Rel 𝐹 ∧ ¬ ∃𝑝𝑞((𝑝𝐹𝑞𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I ) ∘ 2nd ))𝑝)))
10080, 81, 993bitr4ri 304 1 (𝐹 Funs ↔ Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  wrex 3071  Vcvv 3475  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 7973  2nd c2nd 7974  ctxp 34802   Fix cfix 34807   Funs cfuns 34809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 7975  df-2nd 7976  df-txp 34826  df-fix 34831  df-funs 34833
This theorem is referenced by:  elfunsg  34888  dfrecs2  34922  dfrdg4  34923
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