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Theorem dmct 25965
Description: The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
dmct  |-  ( A  ~<_  om  ->  dom  A  ~<_  om )

Proof of Theorem dmct
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dmresv 5291 . 2  |-  dom  ( A  |`  _V )  =  dom  A
2 resss 5129 . . . . 5  |-  ( A  |`  _V )  C_  A
3 ctex 25959 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
4 ssexg 4433 . . . . 5  |-  ( ( ( A  |`  _V )  C_  A  /\  A  e. 
_V )  ->  ( A  |`  _V )  e. 
_V )
52, 3, 4sylancr 663 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  e.  _V )
6 fvex 5696 . . . . . . 7  |-  ( 1st `  x )  e.  _V
7 eqid 2438 . . . . . . 7  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
86, 7fnmpti 5534 . . . . . 6  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  Fn  ( A  |`  _V )
9 dffn4 5621 . . . . . 6  |-  ( ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  Fn  ( A  |`  _V )  <->  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) ) )
108, 9mpbi 208 . . . . 5  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
11 relres 5133 . . . . . 6  |-  Rel  ( A  |`  _V )
12 reldm 6620 . . . . . 6  |-  ( Rel  ( A  |`  _V )  ->  dom  ( A  |`  _V )  =  ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) )
13 foeq3 5613 . . . . . 6  |-  ( dom  ( A  |`  _V )  =  ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )  ->  (
( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )  <->  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) ) ) )
1411, 12, 13mp2b 10 . . . . 5  |-  ( ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )  <->  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) ) )
1510, 14mpbir 209 . . . 4  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )
16 fodomg 8684 . . . 4  |-  ( ( A  |`  _V )  e.  _V  ->  ( (
x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )  ->  dom  ( A  |`  _V )  ~<_  ( A  |` 
_V ) ) )
175, 15, 16mpisyl 18 . . 3  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  ( A  |`  _V ) )
18 ssdomg 7347 . . . . 5  |-  ( A  e.  _V  ->  (
( A  |`  _V )  C_  A  ->  ( A  |` 
_V )  ~<_  A ) )
193, 2, 18mpisyl 18 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  A )
20 domtr 7354 . . . 4  |-  ( ( ( A  |`  _V )  ~<_  A  /\  A  ~<_  om )  ->  ( A  |`  _V )  ~<_  om )
2119, 20mpancom 669 . . 3  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  om )
22 domtr 7354 . . 3  |-  ( ( dom  ( A  |`  _V )  ~<_  ( A  |` 
_V )  /\  ( A  |`  _V )  ~<_  om )  ->  dom  ( A  |`  _V )  ~<_  om )
2317, 21, 22syl2anc 661 . 2  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  om )
241, 23syl5eqbrr 4321 1  |-  ( A  ~<_  om  ->  dom  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   dom cdm 4835   ran crn 4836    |` cres 4837   Rel wrel 4840    Fn wfn 5408   -onto->wfo 5411   ` cfv 5413   omcom 6471   1stc1st 6570    ~<_ 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-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-ac2 8624
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  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-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-recs 6824  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-card 8101  df-acn 8104  df-ac 8278
This theorem is referenced by:  rnct  25967
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