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Theorem cnviinm 4886
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 2116 . . 3 (𝑦 = 𝑎 → (𝑦𝐴𝑎𝐴))
21cbvexv 1811 . 2 (∃𝑦 𝑦𝐴 ↔ ∃𝑎 𝑎𝐴)
3 eleq1 2116 . . . 4 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
43cbvexv 1811 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
5 relcnv 4730 . . . 4 Rel 𝑥𝐴 𝐵
6 r19.2m 3336 . . . . . . . 8 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵 ⊆ (V × V)) → ∃𝑥𝐴 𝐵 ⊆ (V × V))
76expcom 113 . . . . . . 7 (∀𝑥𝐴 𝐵 ⊆ (V × V) → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V)))
8 relcnv 4730 . . . . . . . . 9 Rel 𝐵
9 df-rel 4379 . . . . . . . . 9 (Rel 𝐵𝐵 ⊆ (V × V))
108, 9mpbi 137 . . . . . . . 8 𝐵 ⊆ (V × V)
1110a1i 9 . . . . . . 7 (𝑥𝐴𝐵 ⊆ (V × V))
127, 11mprg 2395 . . . . . 6 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝐵 ⊆ (V × V))
13 iinss 3735 . . . . . 6 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
1412, 13syl 14 . . . . 5 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 ⊆ (V × V))
15 df-rel 4379 . . . . 5 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
1614, 15sylibr 141 . . . 4 (∃𝑥 𝑥𝐴 → Rel 𝑥𝐴 𝐵)
17 vex 2577 . . . . . . . 8 𝑏 ∈ V
18 vex 2577 . . . . . . . 8 𝑎 ∈ V
1917, 18opex 3993 . . . . . . 7 𝑏, 𝑎⟩ ∈ V
20 eliin 3689 . . . . . . 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 4544 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑥𝐴 𝐵)
2318, 17opex 3993 . . . . . . . 8 𝑎, 𝑏⟩ ∈ V
24 eliin 3689 . . . . . . . 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 4544 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
2726ralbii 2347 . . . . . . 7 (∀𝑥𝐴𝑎, 𝑏⟩ ∈ 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2825, 27bitri 177 . . . . . 6 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑏, 𝑎⟩ ∈ 𝐵)
2921, 22, 283bitr4i 205 . . . . 5 (⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑥𝐴 𝐵)
3029eqrelriv 4460 . . . 4 ((Rel 𝑥𝐴 𝐵 ∧ Rel 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
315, 16, 30sylancr 399 . . 3 (∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
324, 31sylbir 129 . 2 (∃𝑎 𝑎𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
332, 32sylbi 118 1 (∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  Vcvv 2574  wss 2944  cop 3405   ciin 3685   × cxp 4370  ccnv 4371  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-iin 3687  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380
This theorem is referenced by: (None)
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