MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvcnv Structured version   Visualization version   GIF version

Theorem cnvcnv 6214
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 6167 . . 3 (𝐴(V × V)) = (𝐴(V × V))
2 cnvin 6167 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
32cnveqi 5888 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
4 relcnv 6125 . . . . . 6 Rel 𝐴
5 df-rel 5696 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
64, 5mpbi 230 . . . . 5 𝐴 ⊆ (V × V)
7 relxp 5707 . . . . . 6 Rel (V × V)
8 dfrel2 6211 . . . . . 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 3982 . . . 4 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
1210, 11mpbi 230 . . 3 𝐴 = (𝐴(V × V))
131, 3, 123eqtr4ri 2774 . 2 𝐴 = (𝐴 ∩ (V × V))
14 relinxp 5827 . . 3 Rel (𝐴 ∩ (V × V))
15 dfrel2 6211 . . 3 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1614, 15mpbi 230 . 2 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1713, 16eqtri 2763 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cin 3962  wss 3963   × cxp 5687  ccnv 5688  Rel wrel 5694
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  cnvcnv2  6215  cnvcnvss  6216  cnvrescnv  6217  structcnvcnv  17187  strfv2d  17236  elcnvcnvintab  43572  relintab  43573  nonrel  43574  elcnvcnvlem  43589  cnvcnvintabd  43590
  Copyright terms: Public domain W3C validator