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Theorem cnvcnv 4959
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 4885 . . . . 5 Rel 𝐴
2 df-rel 4514 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
31, 2mpbi 144 . . . 4 𝐴 ⊆ (V × V)
4 relxp 4616 . . . . 5 Rel (V × V)
5 dfrel2 4957 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 144 . . . 4 (V × V) = (V × V)
73, 6sseqtrri 3100 . . 3 𝐴(V × V)
8 dfss 3053 . . 3 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
97, 8mpbi 144 . 2 𝐴 = (𝐴(V × V))
10 cnvin 4914 . 2 (𝐴(V × V)) = (𝐴(V × V))
11 cnvin 4914 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
1211cnveqi 4682 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
13 inss2 3265 . . . . 5 (𝐴 ∩ (V × V)) ⊆ (V × V)
14 df-rel 4514 . . . . 5 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 145 . . . 4 Rel (𝐴 ∩ (V × V))
16 dfrel2 4957 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1715, 16mpbi 144 . . 3 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1812, 17eqtr3i 2138 . 2 (𝐴(V × V)) = (𝐴 ∩ (V × V))
199, 10, 183eqtr2i 2142 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff set class
Syntax hints:   = wceq 1314  Vcvv 2658  cin 3038  wss 3039   × cxp 4505  ccnv 4506  Rel wrel 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515
This theorem is referenced by:  cnvcnv2  4960  cnvcnvss  4961  structcnvcnv  11870  strslfv2d  11896
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