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Theorem cnviinm 4985
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 eleq1 2151 . . 3 (𝑦 = 𝑎 → (𝑦𝐴𝑎𝐴))
21cbvexv 1844 . 2 (∃𝑦 𝑦𝐴 ↔ ∃𝑎 𝑎𝐴)
3 eleq1 2151 . . . 4 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
43cbvexv 1844 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
5 relcnv 4823 . . . 4 Rel 𝑥𝐴 𝐵
6 r19.2m 3373 . . . . . . . 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 4823 . . . . . . . . 9 Rel 𝐵
9 df-rel 4458 . . . . . . . . 9 (Rel 𝐵𝐵 ⊆ (V × V))
108, 9mpbi 144 . . . . . . . 8 𝐵 ⊆ (V × V)
1110a1i 9 . . . . . . 7 (𝑥𝐴𝐵 ⊆ (V × V))
127, 11mprg 2433 . . . . . 6 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V))
13 iinss 3787 . . . . . 6 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
1412, 13syl 14 . . . . 5 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 ⊆ (V × V))
15 df-rel 4458 . . . . 5 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
1614, 15sylibr 133 . . . 4 (∃𝑥 𝑥𝐴 → Rel 𝑥𝐴 𝐵)
17 vex 2623 . . . . . . . 8 𝑏 ∈ V
18 vex 2623 . . . . . . . 8 𝑎 ∈ V
1917, 18opex 4065 . . . . . . 7 𝑏, 𝑎⟩ ∈ V
20 eliin 3741 . . . . . . 7 (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵))
2119, 20ax-mp 7 . . . . . 6 (⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2218, 17opelcnv 4631 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵)
2318, 17opex 4065 . . . . . . . 8 𝑎, 𝑏⟩ ∈ V
24 eliin 3741 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵))
2523, 24ax-mp 7 . . . . . . 7 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵)
2618, 17opelcnv 4631 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
2726ralbii 2385 . . . . . . 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 4544 . . . 4 ((Rel 𝑥𝐴 𝐵 ∧ Rel 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
315, 16, 30sylancr 406 . . 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 1290  wex 1427  wcel 1439  wral 2360  wrex 2361  Vcvv 2620  wss 3000  cop 3453   ciin 3737   × cxp 4449  ccnv 4450  Rel wrel 4456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-iin 3739  df-br 3852  df-opab 3906  df-xp 4457  df-rel 4458  df-cnv 4459
This theorem is referenced by: (None)
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