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Theorem cnvcnv 6212
Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.)
Assertion
Ref Expression
cnvcnv 𝐴 = (𝐴 ∩ (V × V))

Proof of Theorem cnvcnv
StepHypRef Expression
1 cnvin 6164 . . 3 (𝐴(V × V)) = (𝐴(V × V))
2 cnvin 6164 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
32cnveqi 5885 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
4 relcnv 6122 . . . . . 6 Rel 𝐴
5 df-rel 5692 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
64, 5mpbi 230 . . . . 5 𝐴 ⊆ (V × V)
7 relxp 5703 . . . . . 6 Rel (V × V)
8 dfrel2 6209 . . . . . 6 (Rel (V × V) ↔ (V × V) = (V × V))
97, 8mpbi 230 . . . . 5 (V × V) = (V × V)
106, 9sseqtrri 4033 . . . 4 𝐴(V × V)
11 dfss 3970 . . . 4 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
1210, 11mpbi 230 . . 3 𝐴 = (𝐴(V × V))
131, 3, 123eqtr4ri 2776 . 2 𝐴 = (𝐴 ∩ (V × V))
14 relinxp 5824 . . 3 Rel (𝐴 ∩ (V × V))
15 dfrel2 6209 . . 3 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1614, 15mpbi 230 . 2 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1713, 16eqtri 2765 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cin 3950  wss 3951   × cxp 5683  ccnv 5684  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  cnvcnv2  6213  cnvcnvss  6214  cnvrescnv  6215  structcnvcnv  17190  strfv2d  17238  elcnvcnvintab  43595  relintab  43596  nonrel  43597  elcnvcnvlem  43612  cnvcnvintabd  43613  tposresg  48778
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