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Theorem brdom7disj 9313
Description: An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
brdom7disj.1 𝐴 ∈ V
brdom7disj.2 𝐵 ∈ V
brdom7disj.3 (𝐴𝐵) = ∅
Assertion
Ref Expression
brdom7disj (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem brdom7disj
Dummy variables 𝑔 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdom7disj.2 . . 3 𝐵 ∈ V
21brdom4 9312 . 2 (𝐴𝐵 ↔ ∃𝑔(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
3 incom 3789 . . . . . . . . . . . . . . . . 17 (𝐵𝐴) = (𝐴𝐵)
4 brdom7disj.3 . . . . . . . . . . . . . . . . 17 (𝐴𝐵) = ∅
53, 4eqtri 2643 . . . . . . . . . . . . . . . 16 (𝐵𝐴) = ∅
6 disjne 4000 . . . . . . . . . . . . . . . 16 (((𝐵𝐴) = ∅ ∧ 𝑥𝐵𝑤𝐴) → 𝑥𝑤)
75, 6mp3an1 1408 . . . . . . . . . . . . . . 15 ((𝑥𝐵𝑤𝐴) → 𝑥𝑤)
8 vex 3193 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
9 vex 3193 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 3193 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
11 vex 3193 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
128, 9, 10, 11opthpr 4359 . . . . . . . . . . . . . . 15 (𝑥𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
137, 12syl 17 . . . . . . . . . . . . . 14 ((𝑥𝐵𝑤𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
14 equcom 1942 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧𝑧 = 𝑥)
15 equcom 1942 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤𝑤 = 𝑦)
1614, 15anbi12i 732 . . . . . . . . . . . . . 14 ((𝑥 = 𝑧𝑦 = 𝑤) ↔ (𝑧 = 𝑥𝑤 = 𝑦))
1713, 16syl6rbb 277 . . . . . . . . . . . . 13 ((𝑥𝐵𝑤𝐴) → ((𝑧 = 𝑥𝑤 = 𝑦) ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
18 df-br 4624 . . . . . . . . . . . . . 14 (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
1918a1i 11 . . . . . . . . . . . . 13 ((𝑥𝐵𝑤𝐴) → (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
2017, 19anbi12d 746 . . . . . . . . . . . 12 ((𝑥𝐵𝑤𝐴) → (((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
2120rexbidva 3044 . . . . . . . . . . 11 (𝑥𝐵 → (∃𝑤𝐴 ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ∃𝑤𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
2221rexbidv 3047 . . . . . . . . . 10 (𝑥𝐵 → (∃𝑧𝐵𝑤𝐴 ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ∃𝑧𝐵𝑤𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
23 rexcom 3093 . . . . . . . . . . 11 (∃𝑧𝐵𝑤𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
24 zfpair2 4878 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
25 eqeq1 2625 . . . . . . . . . . . . . 14 (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
2625anbi1d 740 . . . . . . . . . . . . 13 (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
27262rexbidv 3052 . . . . . . . . . . . 12 (𝑣 = {𝑥, 𝑦} → (∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
2824, 27elab 3338 . . . . . . . . . . 11 ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
2923, 28bitr4i 267 . . . . . . . . . 10 (∃𝑧𝐵𝑤𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
3022, 29syl6rbb 277 . . . . . . . . 9 (𝑥𝐵 → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑧𝐵𝑤𝐴 ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝑧𝑔𝑤)))
3130adantr 481 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑧𝐵𝑤𝐴 ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝑧𝑔𝑤)))
32 breq1 4626 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧𝑔𝑤𝑥𝑔𝑤))
33 breq2 4627 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥𝑔𝑤𝑥𝑔𝑦))
3432, 33ceqsrex2v 3326 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → (∃𝑧𝐵𝑤𝐴 ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑥𝑔𝑦))
3531, 34bitrd 268 . . . . . . 7 ((𝑥𝐵𝑦𝐴) → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑥𝑔𝑦))
3635rmobidva 3123 . . . . . 6 (𝑥𝐵 → (∃*𝑦𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃*𝑦𝐴 𝑥𝑔𝑦))
3736ralbiia 2975 . . . . 5 (∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦)
38 zfpair2 4878 . . . . . . . . . . 11 {𝑦, 𝑥} ∈ V
39 eqeq1 2625 . . . . . . . . . . . . 13 (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
4039anbi1d 740 . . . . . . . . . . . 12 (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
41402rexbidv 3052 . . . . . . . . . . 11 (𝑣 = {𝑦, 𝑥} → (∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
4238, 41elab 3338 . . . . . . . . . 10 ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
43 disjne 4000 . . . . . . . . . . . . . . . . 17 (((𝐵𝐴) = ∅ ∧ 𝑧𝐵𝑥𝐴) → 𝑧𝑥)
445, 43mp3an1 1408 . . . . . . . . . . . . . . . 16 ((𝑧𝐵𝑥𝐴) → 𝑧𝑥)
4544ancoms 469 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝑧𝐵) → 𝑧𝑥)
4610, 11, 9, 8opthpr 4359 . . . . . . . . . . . . . . 15 (𝑧𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦𝑤 = 𝑥)))
4745, 46syl 17 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑧𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦𝑤 = 𝑥)))
48 eqcom 2628 . . . . . . . . . . . . . 14 ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
49 ancom 466 . . . . . . . . . . . . . 14 ((𝑤 = 𝑥𝑧 = 𝑦) ↔ (𝑧 = 𝑦𝑤 = 𝑥))
5047, 48, 493bitr4g 303 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥𝑧 = 𝑦)))
5118bicomi 214 . . . . . . . . . . . . . 14 (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧𝑔𝑤)
5251a1i 11 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧𝑔𝑤))
5350, 52anbi12d 746 . . . . . . . . . . . 12 ((𝑥𝐴𝑧𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5453rexbidva 3044 . . . . . . . . . . 11 (𝑥𝐴 → (∃𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5554rexbidv 3047 . . . . . . . . . 10 (𝑥𝐴 → (∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5642, 55syl5bb 272 . . . . . . . . 9 (𝑥𝐴 → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5756adantr 481 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
58 breq2 4627 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑧𝑔𝑤𝑧𝑔𝑥))
59 breq1 4626 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑔𝑥𝑦𝑔𝑥))
6058, 59ceqsrex2v 3326 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
6157, 60bitrd 268 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
6261rexbidva 3044 . . . . . 6 (𝑥𝐴 → (∃𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦𝐵 𝑦𝑔𝑥))
6362ralbiia 2975 . . . . 5 (∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥)
64 brdom7disj.1 . . . . . . 7 𝐴 ∈ V
65 snex 4879 . . . . . . . 8 {{𝑧, 𝑤}} ∈ V
66 simpl 473 . . . . . . . . . 10 ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
6766ss2abi 3659 . . . . . . . . 9 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣𝑣 = {𝑧, 𝑤}}
68 df-sn 4156 . . . . . . . . 9 {{𝑧, 𝑤}} = {𝑣𝑣 = {𝑧, 𝑤}}
6967, 68sseqtr4i 3623 . . . . . . . 8 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
7065, 69ssexi 4773 . . . . . . 7 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
7164, 1, 70ab2rexex2 7120 . . . . . 6 {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
72 eleq2 2687 . . . . . . . . 9 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑥, 𝑦} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7372rmobidv 3124 . . . . . . . 8 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ↔ ∃*𝑦𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7473ralbidv 2982 . . . . . . 7 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ↔ ∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
75 eleq2 2687 . . . . . . . . 9 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑦, 𝑥} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7675rexbidv 3047 . . . . . . . 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)}))
7776ralbidv 2982 . . . . . . 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)}))
7874, 77anbi12d 746 . . . . . 6 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ((∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) ↔ (∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})))
7971, 78spcev 3290 . . . . 5 ((∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}) → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
8037, 63, 79syl2anbr 497 . . . 4 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
8180exlimiv 1855 . . 3 (∃𝑔(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
82 preq1 4245 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
8382eleq1d 2683 . . . . . . . 8 (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
84 preq2 4246 . . . . . . . . 9 (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
8584eleq1d 2683 . . . . . . . 8 (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
86 eqid 2621 . . . . . . . 8 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
878, 9, 83, 85, 86brab 4968 . . . . . . 7 (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
8887rmobii 3126 . . . . . 6 (∃*𝑦𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓)
8988ralbii 2976 . . . . 5 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓)
90 preq1 4245 . . . . . . . . 9 (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
9190eleq1d 2683 . . . . . . . 8 (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
92 preq2 4246 . . . . . . . . 9 (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
9392eleq1d 2683 . . . . . . . 8 (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
949, 8, 91, 93, 86brab 4968 . . . . . . 7 (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
9594rexbii 3036 . . . . . 6 (∃𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓)
9695ralbii 2976 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓)
97 df-opab 4684 . . . . . . 7 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
98 vuniex 6919 . . . . . . . 8 𝑓 ∈ V
9911prid1 4274 . . . . . . . . . . 11 𝑤 ∈ {𝑤, 𝑧}
100 elunii 4414 . . . . . . . . . . 11 ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
10199, 100mpan 705 . . . . . . . . . 10 ({𝑤, 𝑧} ∈ 𝑓𝑤 𝑓)
102101adantl 482 . . . . . . . . 9 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
103102exlimiv 1855 . . . . . . . 8 (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
10410prid2 4275 . . . . . . . . . . 11 𝑧 ∈ {𝑤, 𝑧}
105 elunii 4414 . . . . . . . . . . 11 ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 𝑓)
106104, 105mpan 705 . . . . . . . . . 10 ({𝑤, 𝑧} ∈ 𝑓𝑧 𝑓)
107106adantl 482 . . . . . . . . 9 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 𝑓)
108 df-sn 4156 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩} = {𝑣𝑣 = ⟨𝑤, 𝑧⟩}
109 snex 4879 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩} ∈ V
110108, 109eqeltrri 2695 . . . . . . . . . 10 {𝑣𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
111 simpl 473 . . . . . . . . . . 11 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
112111ss2abi 3659 . . . . . . . . . 10 {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣𝑣 = ⟨𝑤, 𝑧⟩}
113110, 112ssexi 4773 . . . . . . . . 9 {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11498, 107, 113abexex 7111 . . . . . . . 8 {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11598, 103, 114abexex 7111 . . . . . . 7 {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11697, 115eqeltri 2694 . . . . . 6 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
117 breq 4625 . . . . . . . . 9 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
118117rmobidv 3124 . . . . . . . 8 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦𝐴 𝑥𝑔𝑦 ↔ ∃*𝑦𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
119118ralbidv 2982 . . . . . . 7 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ↔ ∀𝑥𝐵 ∃*𝑦𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
120 breq 4625 . . . . . . . . 9 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
121120rexbidv 3047 . . . . . . . 8 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦𝐵 𝑦𝑔𝑥 ↔ ∃𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
122121ralbidv 2982 . . . . . . 7 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
123119, 122anbi12d 746 . . . . . 6 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) ↔ (∀𝑥𝐵 ∃*𝑦𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥)))
124116, 123spcev 3290 . . . . 5 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥) → ∃𝑔(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
12589, 96, 124syl2anbr 497 . . . 4 ((∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
126125exlimiv 1855 . . 3 (∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
12781, 126impbii 199 . 2 (∃𝑔(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
1282, 127bitri 264 1 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2908  wrex 2909  ∃*wrmo 2911  Vcvv 3190  cin 3559  c0 3897  {csn 4155  {cpr 4157  cop 4161   cuni 4409   class class class wbr 4623  {copab 4682  cdom 7913
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-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-ac2 9245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  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 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-card 8725  df-acn 8728  df-ac 8899
This theorem is referenced by: (None)
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