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Theorem cnviinm 5185
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 2250 . . 3 (𝑦 = 𝑎 → (𝑦𝐴𝑎𝐴))
21cbvexv 1930 . 2 (∃𝑦 𝑦𝐴 ↔ ∃𝑎 𝑎𝐴)
3 eleq1w 2250 . . . 4 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
43cbvexv 1930 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
5 relcnv 5021 . . . 4 Rel 𝑥𝐴 𝐵
6 r19.2m 3524 . . . . . . . 8 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵 ⊆ (V × V)) → ∃𝑥𝐴 𝐵 ⊆ (V × V))
76expcom 116 . . . . . . 7 (∀𝑥𝐴 𝐵 ⊆ (V × V) → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V)))
8 relcnv 5021 . . . . . . . . 9 Rel 𝐵
9 df-rel 4648 . . . . . . . . 9 (Rel 𝐵𝐵 ⊆ (V × V))
108, 9mpbi 145 . . . . . . . 8 𝐵 ⊆ (V × V)
1110a1i 9 . . . . . . 7 (𝑥𝐴𝐵 ⊆ (V × V))
127, 11mprg 2547 . . . . . 6 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V))
13 iinss 3953 . . . . . 6 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
1412, 13syl 14 . . . . 5 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 ⊆ (V × V))
15 df-rel 4648 . . . . 5 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
1614, 15sylibr 134 . . . 4 (∃𝑥 𝑥𝐴 → Rel 𝑥𝐴 𝐵)
17 vex 2755 . . . . . . . 8 𝑏 ∈ V
18 vex 2755 . . . . . . . 8 𝑎 ∈ V
1917, 18opex 4244 . . . . . . 7 𝑏, 𝑎⟩ ∈ V
20 eliin 3906 . . . . . . 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 4824 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵)
2318, 17opex 4244 . . . . . . . 8 𝑎, 𝑏⟩ ∈ V
24 eliin 3906 . . . . . . . 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 4824 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
2726ralbii 2496 . . . . . . 7 (∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2825, 27bitri 184 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2921, 22, 283bitr4i 212 . . . . 5 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵)
3029eqrelriv 4734 . . . 4 ((Rel 𝑥𝐴 𝐵 ∧ Rel 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
315, 16, 30sylancr 414 . . 3 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
324, 31sylbir 135 . 2 (∃𝑎 𝑎𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
332, 32sylbi 121 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  Vcvv 2752  wss 3144  cop 3610   ciin 3902   × cxp 4639  ccnv 4640  Rel wrel 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-iin 3904  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649
This theorem is referenced by: (None)
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