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Theorem cnvcnv 4870
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 4797 . . . . 5 Rel 𝐴
2 df-rel 4435 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
31, 2mpbi 143 . . . 4 𝐴 ⊆ (V × V)
4 relxp 4535 . . . . 5 Rel (V × V)
5 dfrel2 4868 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 143 . . . 4 (V × V) = (V × V)
73, 6sseqtr4i 3057 . . 3 𝐴(V × V)
8 dfss 3011 . . 3 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
97, 8mpbi 143 . 2 𝐴 = (𝐴(V × V))
10 cnvin 4826 . 2 (𝐴(V × V)) = (𝐴(V × V))
11 cnvin 4826 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
1211cnveqi 4599 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
13 inss2 3219 . . . . 5 (𝐴 ∩ (V × V)) ⊆ (V × V)
14 df-rel 4435 . . . . 5 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 144 . . . 4 Rel (𝐴 ∩ (V × V))
16 dfrel2 4868 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1715, 16mpbi 143 . . 3 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1812, 17eqtr3i 2110 . 2 (𝐴(V × V)) = (𝐴 ∩ (V × V))
199, 10, 183eqtr2i 2114 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  Vcvv 2619  cin 2996  wss 2997   × cxp 4426  ccnv 4427  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  cnvcnv2  4871  cnvcnvss  4872
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