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Theorem cnvcnv 4800
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 4730 . . . . 5 Rel 𝐴
2 df-rel 4379 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
31, 2mpbi 137 . . . 4 𝐴 ⊆ (V × V)
4 relxp 4474 . . . . 5 Rel (V × V)
5 dfrel2 4798 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 137 . . . 4 (V × V) = (V × V)
73, 6sseqtr4i 3005 . . 3 𝐴(V × V)
8 dfss 2959 . . 3 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
97, 8mpbi 137 . 2 𝐴 = (𝐴(V × V))
10 cnvin 4758 . 2 (𝐴(V × V)) = (𝐴(V × V))
11 cnvin 4758 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
1211cnveqi 4537 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
13 inss2 3185 . . . . 5 (𝐴 ∩ (V × V)) ⊆ (V × V)
14 df-rel 4379 . . . . 5 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 138 . . . 4 Rel (𝐴 ∩ (V × V))
16 dfrel2 4798 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1715, 16mpbi 137 . . 3 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1812, 17eqtr3i 2078 . 2 (𝐴(V × V)) = (𝐴 ∩ (V × V))
199, 10, 183eqtr2i 2082 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  Vcvv 2574  cin 2943  wss 2944   × cxp 4370  ccnv 4371  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380
This theorem is referenced by:  cnvcnv2  4801  cnvcnvss  4802
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