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Theorem cnvcnv 5803
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 5757 . . 3 (𝐴(V × V)) = (𝐴(V × V))
2 cnvin 5757 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
32cnveqi 5500 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
4 relcnv 5720 . . . . . 6 Rel 𝐴
5 df-rel 5319 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
64, 5mpbi 222 . . . . 5 𝐴 ⊆ (V × V)
7 relxp 5330 . . . . . 6 Rel (V × V)
8 dfrel2 5800 . . . . . 6 (Rel (V × V) ↔ (V × V) = (V × V))
97, 8mpbi 222 . . . . 5 (V × V) = (V × V)
106, 9sseqtr4i 3834 . . . 4 𝐴(V × V)
11 dfss 3784 . . . 4 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
1210, 11mpbi 222 . . 3 𝐴 = (𝐴(V × V))
131, 3, 123eqtr4ri 2832 . 2 𝐴 = (𝐴 ∩ (V × V))
14 inss2 4029 . . . 4 (𝐴 ∩ (V × V)) ⊆ (V × V)
15 df-rel 5319 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1614, 15mpbir 223 . . 3 Rel (𝐴 ∩ (V × V))
17 dfrel2 5800 . . 3 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1816, 17mpbi 222 . 2 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1913, 18eqtri 2821 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  Vcvv 3385  cin 3768  wss 3769   × cxp 5310  ccnv 5311  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320
This theorem is referenced by:  cnvcnv2  5804  cnvcnvss  5805  structcnvcnv  16198  strfv2d  16230  elcnvcnvintab  38671  relintab  38672  nonrel  38673  elcnvcnvlem  38688  cnvcnvintabd  38689
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