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Theorem brdom6disj 9392
Description: An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
Hypotheses
Ref Expression
brdom7disj.1 𝐴 ∈ V
brdom7disj.2 𝐵 ∈ V
brdom7disj.3 (𝐴𝐵) = ∅
Assertion
Ref Expression
brdom6disj (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem brdom6disj
Dummy variables 𝑔 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdom7disj.2 . . 3 𝐵 ∈ V
21brdom5 9389 . 2 (𝐴𝐵 ↔ ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
3 zfpair2 4937 . . . . . . . . . 10 {𝑥, 𝑦} ∈ V
4 eqeq1 2655 . . . . . . . . . . . . 13 (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
54anbi1d 741 . . . . . . . . . . . 12 (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
6 df-br 4686 . . . . . . . . . . . . 13 (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
76anbi2i 730 . . . . . . . . . . . 12 (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
85, 7syl6bbr 278 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
982rexbidv 3086 . . . . . . . . . 10 (𝑣 = {𝑥, 𝑦} → (∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
103, 9elab 3382 . . . . . . . . 9 ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤))
11 incom 3838 . . . . . . . . . . . . . . . . 17 (𝐵𝐴) = (𝐴𝐵)
12 brdom7disj.3 . . . . . . . . . . . . . . . . 17 (𝐴𝐵) = ∅
1311, 12eqtri 2673 . . . . . . . . . . . . . . . 16 (𝐵𝐴) = ∅
14 disjne 4055 . . . . . . . . . . . . . . . 16 (((𝐵𝐴) = ∅ ∧ 𝑥𝐵𝑤𝐴) → 𝑥𝑤)
1513, 14mp3an1 1451 . . . . . . . . . . . . . . 15 ((𝑥𝐵𝑤𝐴) → 𝑥𝑤)
16 vex 3234 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
17 vex 3234 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
18 vex 3234 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
19 vex 3234 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
2016, 17, 18, 19opthpr 4415 . . . . . . . . . . . . . . 15 (𝑥𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
2115, 20syl 17 . . . . . . . . . . . . . 14 ((𝑥𝐵𝑤𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
22 breq12 4690 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝑔𝑦𝑧𝑔𝑤))
2322biimprd 238 . . . . . . . . . . . . . 14 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑧𝑔𝑤𝑥𝑔𝑦))
2421, 23syl6bi 243 . . . . . . . . . . . . 13 ((𝑥𝐵𝑤𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} → (𝑧𝑔𝑤𝑥𝑔𝑦)))
2524impd 446 . . . . . . . . . . . 12 ((𝑥𝐵𝑤𝐴) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
2625ex 449 . . . . . . . . . . 11 (𝑥𝐵 → (𝑤𝐴 → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
2726adantrd 483 . . . . . . . . . 10 (𝑥𝐵 → ((𝑤𝐴𝑧𝐵) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
2827rexlimdvv 3066 . . . . . . . . 9 (𝑥𝐵 → (∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
2910, 28syl5bi 232 . . . . . . . 8 (𝑥𝐵 → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → 𝑥𝑔𝑦))
3029alrimiv 1895 . . . . . . 7 (𝑥𝐵 → ∀𝑦({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → 𝑥𝑔𝑦))
31 moim 2548 . . . . . . 7 (∀𝑦({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → 𝑥𝑔𝑦) → (∃*𝑦 𝑥𝑔𝑦 → ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
3230, 31syl 17 . . . . . 6 (𝑥𝐵 → (∃*𝑦 𝑥𝑔𝑦 → ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
3332ralimia 2979 . . . . 5 (∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 → ∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
34 zfpair2 4937 . . . . . . . . . . . 12 {𝑦, 𝑥} ∈ V
35 eqeq1 2655 . . . . . . . . . . . . . 14 (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
3635anbi1d 741 . . . . . . . . . . . . 13 (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
37362rexbidv 3086 . . . . . . . . . . . 12 (𝑣 = {𝑦, 𝑥} → (∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
3834, 37elab 3382 . . . . . . . . . . 11 ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
39 disjne 4055 . . . . . . . . . . . . . . . . . 18 (((𝐵𝐴) = ∅ ∧ 𝑧𝐵𝑥𝐴) → 𝑧𝑥)
4013, 39mp3an1 1451 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑥𝐴) → 𝑧𝑥)
4140ancoms 468 . . . . . . . . . . . . . . . 16 ((𝑥𝐴𝑧𝐵) → 𝑧𝑥)
4218, 19, 17, 16opthpr 4415 . . . . . . . . . . . . . . . 16 (𝑧𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦𝑤 = 𝑥)))
4341, 42syl 17 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝑧𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦𝑤 = 𝑥)))
44 eqcom 2658 . . . . . . . . . . . . . . 15 ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
45 ancom 465 . . . . . . . . . . . . . . 15 ((𝑤 = 𝑥𝑧 = 𝑦) ↔ (𝑧 = 𝑦𝑤 = 𝑥))
4643, 44, 453bitr4g 303 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑧𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥𝑧 = 𝑦)))
476bicomi 214 . . . . . . . . . . . . . . 15 (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧𝑔𝑤)
4847a1i 11 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑧𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧𝑔𝑤))
4946, 48anbi12d 747 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5049rexbidva 3078 . . . . . . . . . . . 12 (𝑥𝐴 → (∃𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5150rexbidv 3081 . . . . . . . . . . 11 (𝑥𝐴 → (∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5238, 51syl5bb 272 . . . . . . . . . 10 (𝑥𝐴 → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5352adantr 480 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
54 breq2 4689 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑧𝑔𝑤𝑧𝑔𝑥))
55 breq1 4688 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧𝑔𝑥𝑦𝑔𝑥))
5654, 55ceqsrex2v 3369 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
5753, 56bitrd 268 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
5857rexbidva 3078 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦𝐵 𝑦𝑔𝑥))
5958ralbiia 3008 . . . . . 6 (∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥)
6059biimpri 218 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥 → ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
61 brdom7disj.1 . . . . . . 7 𝐴 ∈ V
62 snex 4938 . . . . . . . 8 {{𝑧, 𝑤}} ∈ V
63 simpl 472 . . . . . . . . . 10 ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
6463ss2abi 3707 . . . . . . . . 9 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣𝑣 = {𝑧, 𝑤}}
65 df-sn 4211 . . . . . . . . 9 {{𝑧, 𝑤}} = {𝑣𝑣 = {𝑧, 𝑤}}
6664, 65sseqtr4i 3671 . . . . . . . 8 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
6762, 66ssexi 4836 . . . . . . 7 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
6861, 1, 67ab2rexex2 7202 . . . . . 6 {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
69 eleq2 2719 . . . . . . . . 9 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑥, 𝑦} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7069mobidv 2519 . . . . . . . 8 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ↔ ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7170ralbidv 3015 . . . . . . 7 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ↔ ∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
72 eleq2 2719 . . . . . . . . 9 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑦, 𝑥} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7372rexbidv 3081 . . . . . . . 8 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∃𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓 ↔ ∃𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7473ralbidv 3015 . . . . . . 7 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓 ↔ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7571, 74anbi12d 747 . . . . . 6 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ((∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) ↔ (∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})))
7668, 75spcev 3331 . . . . 5 ((∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}) → ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
7733, 60, 76syl2an 493 . . . 4 ((∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
7877exlimiv 1898 . . 3 (∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
79 preq1 4300 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
8079eleq1d 2715 . . . . . . . 8 (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
81 preq2 4301 . . . . . . . . 9 (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
8281eleq1d 2715 . . . . . . . 8 (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
83 eqid 2651 . . . . . . . 8 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
8416, 17, 80, 82, 83brab 5027 . . . . . . 7 (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
8584mobii 2521 . . . . . 6 (∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
8685ralbii 3009 . . . . 5 (∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
87 preq1 4300 . . . . . . . . 9 (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
8887eleq1d 2715 . . . . . . . 8 (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
89 preq2 4301 . . . . . . . . 9 (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
9089eleq1d 2715 . . . . . . . 8 (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
9117, 16, 88, 90, 83brab 5027 . . . . . . 7 (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
9291rexbii 3070 . . . . . 6 (∃𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓)
9392ralbii 3009 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓)
94 df-opab 4746 . . . . . . 7 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
95 vuniex 6996 . . . . . . . 8 𝑓 ∈ V
9619prid1 4329 . . . . . . . . . . 11 𝑤 ∈ {𝑤, 𝑧}
97 elunii 4473 . . . . . . . . . . 11 ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
9896, 97mpan 706 . . . . . . . . . 10 ({𝑤, 𝑧} ∈ 𝑓𝑤 𝑓)
9998adantl 481 . . . . . . . . 9 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
10099exlimiv 1898 . . . . . . . 8 (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
10118prid2 4330 . . . . . . . . . . 11 𝑧 ∈ {𝑤, 𝑧}
102 elunii 4473 . . . . . . . . . . 11 ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 𝑓)
103101, 102mpan 706 . . . . . . . . . 10 ({𝑤, 𝑧} ∈ 𝑓𝑧 𝑓)
104103adantl 481 . . . . . . . . 9 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 𝑓)
105 df-sn 4211 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩} = {𝑣𝑣 = ⟨𝑤, 𝑧⟩}
106 snex 4938 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩} ∈ V
107105, 106eqeltrri 2727 . . . . . . . . . 10 {𝑣𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
108 simpl 472 . . . . . . . . . . 11 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
109108ss2abi 3707 . . . . . . . . . 10 {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣𝑣 = ⟨𝑤, 𝑧⟩}
110107, 109ssexi 4836 . . . . . . . . 9 {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11195, 104, 110abexex 7193 . . . . . . . 8 {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11295, 100, 111abexex 7193 . . . . . . 7 {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11394, 112eqeltri 2726 . . . . . 6 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
114 breq 4687 . . . . . . . . 9 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
115114mobidv 2519 . . . . . . . 8 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦 𝑥𝑔𝑦 ↔ ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
116115ralbidv 3015 . . . . . . 7 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ↔ ∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
117 breq 4687 . . . . . . . . 9 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
118117rexbidv 3081 . . . . . . . 8 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦𝐵 𝑦𝑔𝑥 ↔ ∃𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
119118ralbidv 3015 . . . . . . 7 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
120116, 119anbi12d 747 . . . . . 6 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → ((∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) ↔ (∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥)))
121113, 120spcev 3331 . . . . 5 ((∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥) → ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
12286, 93, 121syl2anbr 496 . . . 4 ((∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
123122exlimiv 1898 . . 3 (∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
12478, 123impbii 199 . 2 (∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
1252, 124bitri 264 1 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃*wmo 2499  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cin 3606  c0 3948  {csn 4210  {cpr 4212  cop 4216   cuni 4468   class class class wbr 4685  {copab 4745  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-acn 8806  df-ac 8977
This theorem is referenced by:  grothprim  9694
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