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Theorem cnvcnv 5490
Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
cnvcnv 𝐴 = (𝐴 ∩ (V × V))

Proof of Theorem cnvcnv
StepHypRef Expression
1 relcnv 5408 . . . . 5 Rel 𝐴
2 df-rel 5034 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
31, 2mpbi 218 . . . 4 𝐴 ⊆ (V × V)
4 relxp 5138 . . . . 5 Rel (V × V)
5 dfrel2 5487 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 218 . . . 4 (V × V) = (V × V)
73, 6sseqtr4i 3600 . . 3 𝐴(V × V)
8 dfss 3554 . . 3 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
97, 8mpbi 218 . 2 𝐴 = (𝐴(V × V))
10 cnvin 5444 . 2 (𝐴(V × V)) = (𝐴(V × V))
11 cnvin 5444 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
1211cnveqi 5206 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
13 inss2 3795 . . . . 5 (𝐴 ∩ (V × V)) ⊆ (V × V)
14 df-rel 5034 . . . . 5 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 219 . . . 4 Rel (𝐴 ∩ (V × V))
16 dfrel2 5487 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1715, 16mpbi 218 . . 3 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1812, 17eqtr3i 2633 . 2 (𝐴(V × V)) = (𝐴 ∩ (V × V))
199, 10, 183eqtr2i 2637 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  Vcvv 3172  cin 3538  wss 3539   × cxp 5025  ccnv 5026  Rel wrel 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035
This theorem is referenced by:  cnvcnv2  5491  cnvcnvss  5492  structcnvcnv  15654  strfv2d  15681  elcnvcnvintab  36690  relintab  36691  nonrel  36692  elcnvcnvlem  36707  cnvcnvintabd  36708
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