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Theorem cnvct 28142
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 5227 . . . 4  |-  Rel  `' A
2 ctex 28135 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
3 cnvexg 6753 . . . . 5  |-  ( A  e.  _V  ->  `' A  e.  _V )
42, 3syl 17 . . . 4  |-  ( A  ~<_  om  ->  `' A  e.  _V )
5 cnven 7652 . . . 4  |-  ( ( Rel  `' A  /\  `' A  e.  _V )  ->  `' A  ~~  `' `' A )
61, 4, 5sylancr 667 . . 3  |-  ( A  ~<_  om  ->  `' A  ~~  `' `' A )
7 cnvcnvss 5310 . . . 4  |-  `' `' A  C_  A
8 ssdomg 7622 . . . 4  |-  ( A  e.  _V  ->  ( `' `' A  C_  A  ->  `' `' A  ~<_  A )
)
92, 7, 8mpisyl 22 . . 3  |-  ( A  ~<_  om  ->  `' `' A  ~<_  A )
10 endomtr 7634 . . 3  |-  ( ( `' A  ~~  `' `' A  /\  `' `' A  ~<_  A )  ->  `' A  ~<_  A )
116, 9, 10syl2anc 665 . 2  |-  ( A  ~<_  om  ->  `' A  ~<_  A )
12 domtr 7629 . 2  |-  ( ( `' A  ~<_  A  /\  A  ~<_  om )  ->  `' A  ~<_  om )
1311, 12mpancom 673 1  |-  ( A  ~<_  om  ->  `' A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   _Vcvv 3087    C_ wss 3442   class class class wbr 4426   `'ccnv 4853   Rel wrel 4859   omcom 6706    ~~ cen 7574    ~<_ cdom 7575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1st 6807  df-2nd 6808  df-en 7578  df-dom 7579
This theorem is referenced by:  rnct  28143
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