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Theorem cnvexg 5208
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 5051 . . 3  |-  Rel  `' A
2 relssdmrn 5193 . . 3  |-  ( Rel  `' A  ->  `' A  C_  ( dom  `' A  X.  ran  `' A ) )
31, 2ax-mp 8 . 2  |-  `' A  C_  ( dom  `' A  X.  ran  `' A )
4 df-rn 4700 . . . 4  |-  ran  A  =  dom  `' A
5 rnexg 4940 . . . 4  |-  ( A  e.  V  ->  ran  A  e.  _V )
64, 5syl5eqelr 2368 . . 3  |-  ( A  e.  V  ->  dom  `' A  e.  _V )
7 dfdm4 4872 . . . 4  |-  dom  A  =  ran  `' A
8 dmexg 4939 . . . 4  |-  ( A  e.  V  ->  dom  A  e.  _V )
97, 8syl5eqelr 2368 . . 3  |-  ( A  e.  V  ->  ran  `' A  e.  _V )
10 xpexg 4800 . . 3  |-  ( ( dom  `' A  e. 
_V  /\  ran  `' A  e.  _V )  ->  ( dom  `' A  X.  ran  `' A )  e.  _V )
116, 9, 10syl2anc 642 . 2  |-  ( A  e.  V  ->  ( dom  `' A  X.  ran  `' A )  e.  _V )
12 ssexg 4160 . 2  |-  ( ( `' A  C_  ( dom  `' A  X.  ran  `' A )  /\  ( dom  `' A  X.  ran  `' A )  e.  _V )  ->  `' A  e. 
_V )
133, 11, 12sylancr 644 1  |-  ( A  e.  V  ->  `' A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    C_ wss 3152    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  cnvex  5209  relcnvexb  5210  cofunex2g  5740  tposexg  6248  cnven  6936  fopwdom  6970  domssex2  7021  domssex  7022  cnvfi  7140  cantnfcl  7368  cantnflt2  7374  cantnflem1  7391  wemapwe  7400  fin1a2lem7  8032  fpwwe  8268  imasle  13425  cnvps  14321  gsumvalx  14451  symginv  14782  itg2gt0  19115  nlfnval  22461  cnvct  23343  orvcval  23658  coinfliprv  23683  relexpcnv  24029  relexprel  24031  injrec2  25119  oriso  25214  dupre1  25243  supwval  25284  intopcoaconb  25540  intopcoaconc  25541  prcnt  25551  pw2f1o2val  27132  lmhmlnmsplit  27185  xpexb  27658  lkrval  29278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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