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Theorem cnvcnv 5077
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 5002 . . . . 5 Rel 𝐴
2 df-rel 4630 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
31, 2mpbi 145 . . . 4 𝐴 ⊆ (V × V)
4 relxp 4732 . . . . 5 Rel (V × V)
5 dfrel2 5075 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 145 . . . 4 (V × V) = (V × V)
73, 6sseqtrri 3190 . . 3 𝐴(V × V)
8 dfss 3143 . . 3 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
97, 8mpbi 145 . 2 𝐴 = (𝐴(V × V))
10 cnvin 5032 . 2 (𝐴(V × V)) = (𝐴(V × V))
11 cnvin 5032 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
1211cnveqi 4798 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
13 inss2 3356 . . . . 5 (𝐴 ∩ (V × V)) ⊆ (V × V)
14 df-rel 4630 . . . . 5 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 146 . . . 4 Rel (𝐴 ∩ (V × V))
16 dfrel2 5075 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1715, 16mpbi 145 . . 3 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1812, 17eqtr3i 2200 . 2 (𝐴(V × V)) = (𝐴 ∩ (V × V))
199, 10, 183eqtr2i 2204 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff set class
Syntax hints:   = wceq 1353  Vcvv 2737  cin 3128  wss 3129   × cxp 4621  ccnv 4622  Rel wrel 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631
This theorem is referenced by:  cnvcnv2  5078  cnvcnvss  5079  structcnvcnv  12458  strslfv2d  12484
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