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Theorem dmct 28298
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 5294 . 2  |-  dom  ( A  |`  _V )  =  dom  A
2 resss 5128 . . . . 5  |-  ( A  |`  _V )  C_  A
3 ctex 7584 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
4 ssexg 4549 . . . . 5  |-  ( ( ( A  |`  _V )  C_  A  /\  A  e. 
_V )  ->  ( A  |`  _V )  e. 
_V )
52, 3, 4sylancr 669 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  e.  _V )
6 fvex 5875 . . . . . . 7  |-  ( 1st `  x )  e.  _V
7 eqid 2451 . . . . . . 7  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
86, 7fnmpti 5706 . . . . . 6  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  Fn  ( A  |`  _V )
9 dffn4 5799 . . . . . 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 212 . . . . 5  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
11 relres 5132 . . . . . 6  |-  Rel  ( A  |`  _V )
12 reldm 6844 . . . . . 6  |-  ( Rel  ( A  |`  _V )  ->  dom  ( A  |`  _V )  =  ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) )
13 foeq3 5791 . . . . . 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 213 . . . 4  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )
16 fodomg 8953 . . . 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 21 . . 3  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  ( A  |`  _V ) )
18 ssdomg 7615 . . . . 5  |-  ( A  e.  _V  ->  (
( A  |`  _V )  C_  A  ->  ( A  |` 
_V )  ~<_  A ) )
193, 2, 18mpisyl 21 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  A )
20 domtr 7622 . . . 4  |-  ( ( ( A  |`  _V )  ~<_  A  /\  A  ~<_  om )  ->  ( A  |`  _V )  ~<_  om )
2119, 20mpancom 675 . . 3  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  om )
22 domtr 7622 . . 3  |-  ( ( dom  ( A  |`  _V )  ~<_  ( A  |` 
_V )  /\  ( A  |`  _V )  ~<_  om )  ->  dom  ( A  |`  _V )  ~<_  om )
2317, 21, 22syl2anc 667 . 2  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  om )
241, 23syl5eqbrr 4437 1  |-  ( A  ~<_  om  ->  dom  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1444    e. wcel 1887   _Vcvv 3045    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   ran crn 4835    |` cres 4836   Rel wrel 4839    Fn wfn 5577   -onto->wfo 5580   ` cfv 5582   omcom 6692   1stc1st 6791    ~<_ cdom 7567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-ac2 8893
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-card 8373  df-acn 8376  df-ac 8547
This theorem is referenced by:  rnct  28300
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