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Theorem cnvct 25983
Description: If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
cnvct  |-  ( A  ~<_  om  ->  `' A  ~<_  om )

Proof of Theorem cnvct
StepHypRef Expression
1 relcnv 5201 . . . 4  |-  Rel  `' A
2 ctex 25976 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
3 cnvexg 6519 . . . . 5  |-  ( A  e.  _V  ->  `' A  e.  _V )
42, 3syl 16 . . . 4  |-  ( A  ~<_  om  ->  `' A  e.  _V )
5 cnven 7377 . . . 4  |-  ( ( Rel  `' A  /\  `' A  e.  _V )  ->  `' A  ~~  `' `' A )
61, 4, 5sylancr 663 . . 3  |-  ( A  ~<_  om  ->  `' A  ~~  `' `' A )
7 cnvcnvss 5287 . . . 4  |-  `' `' A  C_  A
8 ssdomg 7347 . . . 4  |-  ( A  e.  _V  ->  ( `' `' A  C_  A  ->  `' `' A  ~<_  A )
)
92, 7, 8mpisyl 18 . . 3  |-  ( A  ~<_  om  ->  `' `' A  ~<_  A )
10 endomtr 7359 . . 3  |-  ( ( `' A  ~~  `' `' A  /\  `' `' A  ~<_  A )  ->  `' A  ~<_  A )
116, 9, 10syl2anc 661 . 2  |-  ( A  ~<_  om  ->  `' A  ~<_  A )
12 domtr 7354 . 2  |-  ( ( `' A  ~<_  A  /\  A  ~<_  om )  ->  `' A  ~<_  om )
1311, 12mpancom 669 1  |-  ( A  ~<_  om  ->  `' A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   _Vcvv 2967    C_ wss 3323   class class class wbr 4287   `'ccnv 4834   Rel wrel 4840   omcom 6471    ~~ cen 7299    ~<_ cdom 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1st 6572  df-2nd 6573  df-en 7303  df-dom 7304
This theorem is referenced by:  rnct  25984
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