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Theorem cnvexg 5114
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 4958 . . 3  |-  Rel  `' A
2 relssdmrn 5099 . . 3  |-  ( Rel  `' A  ->  `' A  C_  ( dom  `'  A  X.  ran  `'  A ) )
31, 2ax-mp 10 . 2  |-  `' A  C_  ( dom  `'  A  X.  ran  `'  A )
4 df-rn 4599 . . . 4  |-  ran  A  =  dom  `'  A
5 rnexg 4847 . . . 4  |-  ( A  e.  V  ->  ran  A  e.  _V )
64, 5syl5eqelr 2338 . . 3  |-  ( A  e.  V  ->  dom  `'  A  e.  _V )
7 dfdm4 4779 . . . 4  |-  dom  A  =  ran  `'  A
8 dmexg 4846 . . . 4  |-  ( A  e.  V  ->  dom  A  e.  _V )
97, 8syl5eqelr 2338 . . 3  |-  ( A  e.  V  ->  ran  `'  A  e.  _V )
10 xpexg 4707 . . 3  |-  ( ( dom  `'  A  e. 
_V  /\  ran  `'  A  e.  _V )  ->  ( dom  `'  A  X.  ran  `'  A )  e.  _V )
116, 9, 10syl2anc 645 . 2  |-  ( A  e.  V  ->  ( dom  `'  A  X.  ran  `'  A )  e.  _V )
12 ssexg 4057 . 2  |-  ( ( `' A  C_  ( dom  `'  A  X.  ran  `'  A )  /\  ( dom  `'  A  X.  ran  `'  A )  e.  _V )  ->  `' A  e. 
_V )
133, 11, 12sylancr 647 1  |-  ( A  e.  V  ->  `' A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   _Vcvv 2727    C_ wss 3078    X. cxp 4578   `'ccnv 4579   dom cdm 4580   ran crn 4581   Rel wrel 4585
This theorem is referenced by:  cnvex  5115  relcnvexb  5116  cofunex2g  5592  tposexg  6100  cnven  6821  fopwdom  6855  domssex2  6906  domssex  6907  cnvfi  7025  cantnfcl  7252  cantnflt2  7258  cantnflem1  7275  wemapwe  7284  fin1a2lem7  7916  fpwwe  8148  imasle  13299  cnvps  14156  gsumvalx  14286  symginv  14617  itg2gt0  18947  nlfnval  22291  relexpcnv  23200  relexprel  23202  injrec2  24285  oriso  24380  dupre1  24409  supwval  24450  intopcoaconb  24706  intopcoaconc  24707  prcnt  24717  pw2f1o2val  26298  lmhmlnmsplit  26351  xpexb  26825  lkrval  27967
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599
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