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Theorem cnvcnv 4716
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 4646 . . . . 5 Rel A
2 df-rel 4295 . . . . 5 (Rel AA ⊆ (V × V))
31, 2mpbi 133 . . . 4 A ⊆ (V × V)
4 relxp 4390 . . . . 5 Rel (V × V)
5 dfrel2 4714 . . . . 5 (Rel (V × V) ↔ (V × V) = (V × V))
64, 5mpbi 133 . . . 4 (V × V) = (V × V)
73, 6sseqtr4i 2972 . . 3 A(V × V)
8 dfss 2926 . . 3 (A(V × V) ↔ A = (A(V × V)))
97, 8mpbi 133 . 2 A = (A(V × V))
10 cnvin 4674 . 2 (A(V × V)) = (A(V × V))
11 cnvin 4674 . . . 4 (A ∩ (V × V)) = (A(V × V))
1211cnveqi 4453 . . 3 (A ∩ (V × V)) = (A(V × V))
13 inss2 3152 . . . . 5 (A ∩ (V × V)) ⊆ (V × V)
14 df-rel 4295 . . . . 5 (Rel (A ∩ (V × V)) ↔ (A ∩ (V × V)) ⊆ (V × V))
1513, 14mpbir 134 . . . 4 Rel (A ∩ (V × V))
16 dfrel2 4714 . . . 4 (Rel (A ∩ (V × V)) ↔ (A ∩ (V × V)) = (A ∩ (V × V)))
1715, 16mpbi 133 . . 3 (A ∩ (V × V)) = (A ∩ (V × V))
1812, 17eqtr3i 2059 . 2 (A(V × V)) = (A ∩ (V × V))
199, 10, 183eqtr2i 2063 1 A = (A ∩ (V × V))
Colors of variables: wff set class
Syntax hints:   = wceq 1242  Vcvv 2551  cin 2910  wss 2911   × cxp 4286  ccnv 4287  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  cnvcnv2  4717  cnvcnvss  4718
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