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Theorem cnvexg 5195
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 5039 . . 3  |-  Rel  `' A
2 relssdmrn 5180 . . 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 4680 . . . 4  |-  ran  A  =  dom  `'  A
5 rnexg 4928 . . . 4  |-  ( A  e.  V  ->  ran  A  e.  _V )
64, 5syl5eqelr 2343 . . 3  |-  ( A  e.  V  ->  dom  `'  A  e.  _V )
7 dfdm4 4860 . . . 4  |-  dom  A  =  ran  `'  A
8 dmexg 4927 . . . 4  |-  ( A  e.  V  ->  dom  A  e.  _V )
97, 8syl5eqelr 2343 . . 3  |-  ( A  e.  V  ->  ran  `'  A  e.  _V )
10 xpexg 4788 . . 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 4134 . 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 2763    C_ wss 3127    X. cxp 4659   `'ccnv 4660   dom cdm 4661   ran crn 4662   Rel wrel 4666
This theorem is referenced by:  cnvex  5196  relcnvexb  5197  cofunex2g  5674  tposexg  6182  cnven  6904  fopwdom  6938  domssex2  6989  domssex  6990  cnvfi  7108  cantnfcl  7336  cantnflt2  7342  cantnflem1  7359  wemapwe  7368  fin1a2lem7  8000  fpwwe  8236  imasle  13387  cnvps  14283  gsumvalx  14413  symginv  14744  itg2gt0  19077  nlfnval  22421  relexpcnv  23401  relexprel  23403  injrec2  24486  oriso  24581  dupre1  24610  supwval  24651  intopcoaconb  24907  intopcoaconc  24908  prcnt  24918  pw2f1o2val  26499  lmhmlnmsplit  26552  xpexb  27026  lkrval  28445
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677  df-dm 4679  df-rn 4680
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