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Theorem cnvcnv 6044
 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 5998 . . 3 (𝐴(V × V)) = (𝐴(V × V))
2 cnvin 5998 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
32cnveqi 5740 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
4 relcnv 5962 . . . . . 6 Rel 𝐴
5 df-rel 5557 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
64, 5mpbi 232 . . . . 5 𝐴 ⊆ (V × V)
7 relxp 5568 . . . . . 6 Rel (V × V)
8 dfrel2 6041 . . . . . 6 (Rel (V × V) ↔ (V × V) = (V × V))
97, 8mpbi 232 . . . . 5 (V × V) = (V × V)
106, 9sseqtrri 4004 . . . 4 𝐴(V × V)
11 dfss 3953 . . . 4 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
1210, 11mpbi 232 . . 3 𝐴 = (𝐴(V × V))
131, 3, 123eqtr4ri 2855 . 2 𝐴 = (𝐴 ∩ (V × V))
14 relinxp 5682 . . 3 Rel (𝐴 ∩ (V × V))
15 dfrel2 6041 . . 3 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1614, 15mpbi 232 . 2 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1713, 16eqtri 2844 1 𝐴 = (𝐴 ∩ (V × V))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1533  Vcvv 3495   ∩ cin 3935   ⊆ wss 3936   × cxp 5548  ◡ccnv 5549  Rel wrel 5555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-rel 5557  df-cnv 5558 This theorem is referenced by:  cnvcnv2  6045  cnvcnvss  6046  cnvrescnv  6047  structcnvcnv  16491  strfv2d  16523  elcnvcnvintab  39935  relintab  39936  nonrel  39937  elcnvcnvlem  39952  cnvcnvintabd  39953
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