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Theorem cnvexg 5206
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 5050 . . 3  |-  Rel  `' A
2 relssdmrn 5191 . . 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 4699 . . . 4  |-  ran  A  =  dom  `'  A
5 rnexg 4939 . . . 4  |-  ( A  e.  V  ->  ran  A  e.  _V )
64, 5syl5eqelr 2369 . . 3  |-  ( A  e.  V  ->  dom  `'  A  e.  _V )
7 dfdm4 4871 . . . 4  |-  dom  A  =  ran  `'  A
8 dmexg 4938 . . . 4  |-  ( A  e.  V  ->  dom  A  e.  _V )
97, 8syl5eqelr 2369 . . 3  |-  ( A  e.  V  ->  ran  `'  A  e.  _V )
10 xpexg 4799 . . 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 4161 . 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 1688   _Vcvv 2789    C_ wss 3153    X. cxp 4686   `'ccnv 4687   dom cdm 4688   ran crn 4689   Rel wrel 4693
This theorem is referenced by:  cnvex  5207  relcnvexb  5208  cofunex2g  5701  tposexg  6209  cnven  6931  fopwdom  6965  domssex2  7016  domssex  7017  cnvfi  7135  cantnfcl  7363  cantnflt2  7369  cantnflem1  7386  wemapwe  7395  fin1a2lem7  8027  fpwwe  8263  imasle  13419  cnvps  14315  gsumvalx  14445  symginv  14776  itg2gt0  19109  nlfnval  22453  relexpcnv  23433  relexprel  23435  injrec2  24518  oriso  24613  dupre1  24642  supwval  24683  intopcoaconb  24939  intopcoaconc  24940  prcnt  24950  pw2f1o2val  26531  lmhmlnmsplit  26584  xpexb  27057  lkrval  28545
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699
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