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Theorem cnvexg 4963
Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
Assertion
Ref Expression
cnvexg  |-  ( A  e.  V  ->  `' A  e.  _V )

Proof of Theorem cnvexg
StepHypRef Expression
1 relcnv 4805 . . 3  |-  Rel  `' A
2 relssdmrn 4946 . . 3  |-  ( Rel  `' A  ->  `' A  C_  ( dom  `' A  X.  ran  `' A ) )
31, 2ax-mp 7 . 2  |-  `' A  C_  ( dom  `' A  X.  ran  `' A )
4 df-rn 4447 . . . 4  |-  ran  A  =  dom  `' A
5 rnexg 4694 . . . 4  |-  ( A  e.  V  ->  ran  A  e.  _V )
64, 5syl5eqelr 2175 . . 3  |-  ( A  e.  V  ->  dom  `' A  e.  _V )
7 dfdm4 4624 . . . 4  |-  dom  A  =  ran  `' A
8 dmexg 4693 . . . 4  |-  ( A  e.  V  ->  dom  A  e.  _V )
97, 8syl5eqelr 2175 . . 3  |-  ( A  e.  V  ->  ran  `' A  e.  _V )
10 xpexg 4548 . . 3  |-  ( ( dom  `' A  e. 
_V  /\  ran  `' A  e.  _V )  ->  ( dom  `' A  X.  ran  `' A )  e.  _V )
116, 9, 10syl2anc 403 . 2  |-  ( A  e.  V  ->  ( dom  `' A  X.  ran  `' A )  e.  _V )
12 ssexg 3976 . 2  |-  ( ( `' A  C_  ( dom  `' A  X.  ran  `' A )  /\  ( dom  `' A  X.  ran  `' A )  e.  _V )  ->  `' A  e. 
_V )
133, 11, 12sylancr 405 1  |-  ( A  e.  V  ->  `' A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   _Vcvv 2619    C_ wss 2999    X. cxp 4434   `'ccnv 4435   dom cdm 4436   ran crn 4437   Rel wrel 4441
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
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 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444  df-dm 4446  df-rn 4447
This theorem is referenced by:  cnvex  4964  relcnvexb  4965  cofunex2g  5875  cnvf1o  5982  brtpos2  6008  tposexg  6015  cnven  6515  cnvct  6516  fopwdom  6542  relcnvfi  6640
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