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Theorem cnvct 26165
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 5313 . . . 4  |-  Rel  `' A
2 ctex 26158 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
3 cnvexg 6633 . . . . 5  |-  ( A  e.  _V  ->  `' A  e.  _V )
42, 3syl 16 . . . 4  |-  ( A  ~<_  om  ->  `' A  e.  _V )
5 cnven 7494 . . . 4  |-  ( ( Rel  `' A  /\  `' A  e.  _V )  ->  `' A  ~~  `' `' A )
61, 4, 5sylancr 663 . . 3  |-  ( A  ~<_  om  ->  `' A  ~~  `' `' A )
7 cnvcnvss 5399 . . . 4  |-  `' `' A  C_  A
8 ssdomg 7464 . . . 4  |-  ( A  e.  _V  ->  ( `' `' A  C_  A  ->  `' `' A  ~<_  A )
)
92, 7, 8mpisyl 18 . . 3  |-  ( A  ~<_  om  ->  `' `' A  ~<_  A )
10 endomtr 7476 . . 3  |-  ( ( `' A  ~~  `' `' A  /\  `' `' A  ~<_  A )  ->  `' A  ~<_  A )
116, 9, 10syl2anc 661 . 2  |-  ( A  ~<_  om  ->  `' A  ~<_  A )
12 domtr 7471 . 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 1758   _Vcvv 3076    C_ wss 3435   class class class wbr 4399   `'ccnv 4946   Rel wrel 4952   omcom 6585    ~~ cen 7416    ~<_ cdom 7417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-1st 6686  df-2nd 6687  df-en 7420  df-dom 7421
This theorem is referenced by:  rnct  26166
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