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Theorem cnvcnv 5505
 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 5422 . . . . 5 Rel 𝐴
2 df-rel 5045 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
31, 2mpbi 219 . . . 4 𝐴 ⊆ (V × V)
4 relxp 5150 . . . . 5 Rel (V × V)
5 dfrel2 5502 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 219 . . . 4 (V × V) = (V × V)
73, 6sseqtr4i 3601 . . 3 𝐴(V × V)
8 dfss 3555 . . 3 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
97, 8mpbi 219 . 2 𝐴 = (𝐴(V × V))
10 cnvin 5459 . 2 (𝐴(V × V)) = (𝐴(V × V))
11 cnvin 5459 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
1211cnveqi 5219 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
13 inss2 3796 . . . . 5 (𝐴 ∩ (V × V)) ⊆ (V × V)
14 df-rel 5045 . . . . 5 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 220 . . . 4 Rel (𝐴 ∩ (V × V))
16 dfrel2 5502 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1715, 16mpbi 219 . . 3 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1812, 17eqtr3i 2634 . 2 (𝐴(V × V)) = (𝐴 ∩ (V × V))
199, 10, 183eqtr2i 2638 1 𝐴 = (𝐴 ∩ (V × V))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   × cxp 5036  ◡ccnv 5037  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  cnvcnv2  5506  cnvcnvss  5507  structcnvcnv  15706  strfv2d  15733  elcnvcnvintab  36907  relintab  36908  nonrel  36909  elcnvcnvlem  36924  cnvcnvintabd  36925
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