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Theorem cnviinm 5050
Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
cnviinm (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem cnviinm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1w 2178 . . 3 (𝑦 = 𝑎 → (𝑦𝐴𝑎𝐴))
21cbvexv 1872 . 2 (∃𝑦 𝑦𝐴 ↔ ∃𝑎 𝑎𝐴)
3 eleq1w 2178 . . . 4 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
43cbvexv 1872 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
5 relcnv 4887 . . . 4 Rel 𝑥𝐴 𝐵
6 r19.2m 3419 . . . . . . . 8 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵 ⊆ (V × V)) → ∃𝑥𝐴 𝐵 ⊆ (V × V))
76expcom 115 . . . . . . 7 (∀𝑥𝐴 𝐵 ⊆ (V × V) → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V)))
8 relcnv 4887 . . . . . . . . 9 Rel 𝐵
9 df-rel 4516 . . . . . . . . 9 (Rel 𝐵𝐵 ⊆ (V × V))
108, 9mpbi 144 . . . . . . . 8 𝐵 ⊆ (V × V)
1110a1i 9 . . . . . . 7 (𝑥𝐴𝐵 ⊆ (V × V))
127, 11mprg 2466 . . . . . 6 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V))
13 iinss 3834 . . . . . 6 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
1412, 13syl 14 . . . . 5 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 ⊆ (V × V))
15 df-rel 4516 . . . . 5 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
1614, 15sylibr 133 . . . 4 (∃𝑥 𝑥𝐴 → Rel 𝑥𝐴 𝐵)
17 vex 2663 . . . . . . . 8 𝑏 ∈ V
18 vex 2663 . . . . . . . 8 𝑎 ∈ V
1917, 18opex 4121 . . . . . . 7 𝑏, 𝑎⟩ ∈ V
20 eliin 3788 . . . . . . 7 (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵))
2119, 20ax-mp 5 . . . . . 6 (⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2218, 17opelcnv 4691 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵)
2318, 17opex 4121 . . . . . . . 8 𝑎, 𝑏⟩ ∈ V
24 eliin 3788 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵))
2523, 24ax-mp 5 . . . . . . 7 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵)
2618, 17opelcnv 4691 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
2726ralbii 2418 . . . . . . 7 (∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2825, 27bitri 183 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2921, 22, 283bitr4i 211 . . . . 5 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵)
3029eqrelriv 4602 . . . 4 ((Rel 𝑥𝐴 𝐵 ∧ Rel 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
315, 16, 30sylancr 410 . . 3 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
324, 31sylbir 134 . 2 (∃𝑎 𝑎𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
332, 32sylbi 120 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wex 1453  wcel 1465  wral 2393  wrex 2394  Vcvv 2660  wss 3041  cop 3500   ciin 3784   × cxp 4507  ccnv 4508  Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iin 3786  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by: (None)
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