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Theorem brdom6disj 10569
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 10566 . 2 (𝐴𝐵 ↔ ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
3 zfpair2 5438 . . . . . . . . 9 {𝑥, 𝑦} ∈ V
4 eqeq1 2738 . . . . . . . . . . . 12 (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
54anbi1d 631 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
6 df-br 5148 . . . . . . . . . . . 12 (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
76anbi2i 623 . . . . . . . . . . 11 (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
85, 7bitr4di 289 . . . . . . . . . 10 (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
982rexbidv 3219 . . . . . . . . 9 (𝑣 = {𝑥, 𝑦} → (∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
103, 9elab 3680 . . . . . . . 8 ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤))
11 incom 4216 . . . . . . . . . . . . . . . 16 (𝐵𝐴) = (𝐴𝐵)
12 brdom7disj.3 . . . . . . . . . . . . . . . 16 (𝐴𝐵) = ∅
1311, 12eqtri 2762 . . . . . . . . . . . . . . 15 (𝐵𝐴) = ∅
14 disjne 4460 . . . . . . . . . . . . . . 15 (((𝐵𝐴) = ∅ ∧ 𝑥𝐵𝑤𝐴) → 𝑥𝑤)
1513, 14mp3an1 1447 . . . . . . . . . . . . . 14 ((𝑥𝐵𝑤𝐴) → 𝑥𝑤)
16 vex 3481 . . . . . . . . . . . . . . 15 𝑥 ∈ V
17 vex 3481 . . . . . . . . . . . . . . 15 𝑦 ∈ V
18 vex 3481 . . . . . . . . . . . . . . 15 𝑧 ∈ V
19 vex 3481 . . . . . . . . . . . . . . 15 𝑤 ∈ V
2016, 17, 18, 19opthpr 4855 . . . . . . . . . . . . . 14 (𝑥𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
2115, 20syl 17 . . . . . . . . . . . . 13 ((𝑥𝐵𝑤𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
22 breq12 5152 . . . . . . . . . . . . . 14 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝑔𝑦𝑧𝑔𝑤))
2322biimprd 248 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑧𝑔𝑤𝑥𝑔𝑦))
2421, 23biimtrdi 253 . . . . . . . . . . . 12 ((𝑥𝐵𝑤𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} → (𝑧𝑔𝑤𝑥𝑔𝑦)))
2524impd 410 . . . . . . . . . . 11 ((𝑥𝐵𝑤𝐴) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
2625ex 412 . . . . . . . . . 10 (𝑥𝐵 → (𝑤𝐴 → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
2726adantrd 491 . . . . . . . . 9 (𝑥𝐵 → ((𝑤𝐴𝑧𝐵) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
2827rexlimdvv 3209 . . . . . . . 8 (𝑥𝐵 → (∃𝑤𝐴𝑧𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
2910, 28biimtrid 242 . . . . . . 7 (𝑥𝐵 → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → 𝑥𝑔𝑦))
3029moimdv 2543 . . . . . 6 (𝑥𝐵 → (∃*𝑦 𝑥𝑔𝑦 → ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
3130ralimia 3077 . . . . 5 (∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 → ∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
32 zfpair2 5438 . . . . . . . . . . . 12 {𝑦, 𝑥} ∈ V
33 eqeq1 2738 . . . . . . . . . . . . . 14 (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
3433anbi1d 631 . . . . . . . . . . . . 13 (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
35342rexbidv 3219 . . . . . . . . . . . 12 (𝑣 = {𝑦, 𝑥} → (∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
3632, 35elab 3680 . . . . . . . . . . 11 ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
37 disjne 4460 . . . . . . . . . . . . . . . . . 18 (((𝐵𝐴) = ∅ ∧ 𝑧𝐵𝑥𝐴) → 𝑧𝑥)
3813, 37mp3an1 1447 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑥𝐴) → 𝑧𝑥)
3938ancoms 458 . . . . . . . . . . . . . . . 16 ((𝑥𝐴𝑧𝐵) → 𝑧𝑥)
4018, 19, 17, 16opthpr 4855 . . . . . . . . . . . . . . . 16 (𝑧𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦𝑤 = 𝑥)))
4139, 40syl 17 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝑧𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦𝑤 = 𝑥)))
42 eqcom 2741 . . . . . . . . . . . . . . 15 ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
43 ancom 460 . . . . . . . . . . . . . . 15 ((𝑤 = 𝑥𝑧 = 𝑦) ↔ (𝑧 = 𝑦𝑤 = 𝑥))
4441, 42, 433bitr4g 314 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑧𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥𝑧 = 𝑦)))
456bicomi 224 . . . . . . . . . . . . . . 15 (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧𝑔𝑤)
4645a1i 11 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑧𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔𝑧𝑔𝑤))
4744, 46anbi12d 632 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
4847rexbidva 3174 . . . . . . . . . . . 12 (𝑥𝐴 → (∃𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
4948rexbidv 3176 . . . . . . . . . . 11 (𝑥𝐴 → (∃𝑤𝐴𝑧𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5036, 49bitrid 283 . . . . . . . . . 10 (𝑥𝐴 → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
5150adantr 480 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
52 breq2 5151 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑧𝑔𝑤𝑧𝑔𝑥))
53 breq1 5150 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧𝑔𝑥𝑦𝑔𝑥))
5452, 53ceqsrex2v 3657 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → (∃𝑤𝐴𝑧𝐵 ((𝑤 = 𝑥𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
5551, 54bitrd 279 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
5655rexbidva 3174 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦𝐵 𝑦𝑔𝑥))
5756ralbiia 3088 . . . . . 6 (∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥)
5857biimpri 228 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥 → ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
59 brdom7disj.1 . . . . . . 7 𝐴 ∈ V
60 snex 5441 . . . . . . . 8 {{𝑧, 𝑤}} ∈ V
61 simpl 482 . . . . . . . . . 10 ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
6261ss2abi 4076 . . . . . . . . 9 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣𝑣 = {𝑧, 𝑤}}
63 df-sn 4631 . . . . . . . . 9 {{𝑧, 𝑤}} = {𝑣𝑣 = {𝑧, 𝑤}}
6462, 63sseqtrri 4032 . . . . . . . 8 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
6560, 64ssexi 5327 . . . . . . 7 {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
6659, 1, 65ab2rexex2 8003 . . . . . 6 {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
67 eleq2 2827 . . . . . . . . 9 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑥, 𝑦} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
6867mobidv 2546 . . . . . . . 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)}))
6968ralbidv 3175 . . . . . . 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)}))
70 eleq2 2827 . . . . . . . . 9 (𝑓 = {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑦, 𝑥} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤𝐴𝑧𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
7170rexbidv 3176 . . . . . . . 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)}))
7271ralbidv 3175 . . . . . . 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)}))
7369, 72anbi12d 632 . . . . . 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)})))
7466, 73spcev 3605 . . . . 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))
7531, 58, 74syl2an 596 . . . 4 ((∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
7675exlimiv 1927 . . 3 (∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
77 preq1 4737 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
7877eleq1d 2823 . . . . . . . 8 (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
79 preq2 4738 . . . . . . . . 9 (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
8079eleq1d 2823 . . . . . . . 8 (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
81 eqid 2734 . . . . . . . 8 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
8216, 17, 78, 80, 81brab 5552 . . . . . . 7 (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
8382mobii 2545 . . . . . 6 (∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
8483ralbii 3090 . . . . 5 (∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
85 preq1 4737 . . . . . . . . 9 (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
8685eleq1d 2823 . . . . . . . 8 (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
87 preq2 4738 . . . . . . . . 9 (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
8887eleq1d 2823 . . . . . . . 8 (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
8917, 16, 86, 88, 81brab 5552 . . . . . . 7 (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
9089rexbii 3091 . . . . . 6 (∃𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓)
9190ralbii 3090 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓)
92 df-opab 5210 . . . . . . 7 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
93 vuniex 7757 . . . . . . . 8 𝑓 ∈ V
9419prid1 4766 . . . . . . . . . . 11 𝑤 ∈ {𝑤, 𝑧}
95 elunii 4916 . . . . . . . . . . 11 ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
9694, 95mpan 690 . . . . . . . . . 10 ({𝑤, 𝑧} ∈ 𝑓𝑤 𝑓)
9796adantl 481 . . . . . . . . 9 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
9897exlimiv 1927 . . . . . . . 8 (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 𝑓)
9918prid2 4767 . . . . . . . . . . 11 𝑧 ∈ {𝑤, 𝑧}
100 elunii 4916 . . . . . . . . . . 11 ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 𝑓)
10199, 100mpan 690 . . . . . . . . . 10 ({𝑤, 𝑧} ∈ 𝑓𝑧 𝑓)
102101adantl 481 . . . . . . . . 9 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 𝑓)
103 df-sn 4631 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩} = {𝑣𝑣 = ⟨𝑤, 𝑧⟩}
104 snex 5441 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩} ∈ V
105103, 104eqeltrri 2835 . . . . . . . . . 10 {𝑣𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
106 simpl 482 . . . . . . . . . . 11 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
107106ss2abi 4076 . . . . . . . . . 10 {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣𝑣 = ⟨𝑤, 𝑧⟩}
108105, 107ssexi 5327 . . . . . . . . 9 {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
10993, 102, 108abexex 7994 . . . . . . . 8 {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11093, 98, 109abexex 7994 . . . . . . 7 {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
11192, 110eqeltri 2834 . . . . . 6 {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
112 breq 5149 . . . . . . . . 9 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
113112mobidv 2546 . . . . . . . 8 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦 𝑥𝑔𝑦 ↔ ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
114113ralbidv 3175 . . . . . . 7 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ↔ ∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
115 breq 5149 . . . . . . . . 9 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
116115rexbidv 3176 . . . . . . . 8 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦𝐵 𝑦𝑔𝑥 ↔ ∃𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
117116ralbidv 3175 . . . . . . 7 (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥 ↔ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
118114, 117anbi12d 632 . . . . . 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)))
119111, 118spcev 3605 . . . . 5 ((∀𝑥𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥) → ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
12084, 91, 119syl2anbr 599 . . . 4 ((∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
121120exlimiv 1927 . . 3 (∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥))
12276, 121impbii 209 . 2 (∃𝑔(∀𝑥𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑔𝑥) ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
1232, 122bitri 275 1 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  ∃*wmo 2535  {cab 2711  wne 2937  wral 3058  wrex 3067  Vcvv 3477  cin 3961  c0 4338  {csn 4630  {cpr 4632  cop 4636   cuni 4911   class class class wbr 5147  {copab 5209  cdom 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-ac2 10500
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-card 9976  df-acn 9979  df-ac 10153
This theorem is referenced by:  grothprim  10871
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