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Theorem dmct 28373
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 5301 . 2  |-  dom  ( A  |`  _V )  =  dom  A
2 resss 5134 . . . . 5  |-  ( A  |`  _V )  C_  A
3 ctex 7602 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
4 ssexg 4542 . . . . 5  |-  ( ( ( A  |`  _V )  C_  A  /\  A  e. 
_V )  ->  ( A  |`  _V )  e. 
_V )
52, 3, 4sylancr 676 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  e.  _V )
6 fvex 5889 . . . . . . 7  |-  ( 1st `  x )  e.  _V
7 eqid 2471 . . . . . . 7  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
86, 7fnmpti 5716 . . . . . 6  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  Fn  ( A  |`  _V )
9 dffn4 5812 . . . . . 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 213 . . . . 5  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
11 relres 5138 . . . . . 6  |-  Rel  ( A  |`  _V )
12 reldm 6863 . . . . . 6  |-  ( Rel  ( A  |`  _V )  ->  dom  ( A  |`  _V )  =  ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) )
13 foeq3 5804 . . . . . 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 214 . . . 4  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )
16 fodomg 8971 . . . 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 7633 . . . . 5  |-  ( A  e.  _V  ->  (
( A  |`  _V )  C_  A  ->  ( A  |` 
_V )  ~<_  A ) )
193, 2, 18mpisyl 21 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  A )
20 domtr 7640 . . . 4  |-  ( ( ( A  |`  _V )  ~<_  A  /\  A  ~<_  om )  ->  ( A  |`  _V )  ~<_  om )
2119, 20mpancom 682 . . 3  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  om )
22 domtr 7640 . . 3  |-  ( ( dom  ( A  |`  _V )  ~<_  ( A  |` 
_V )  /\  ( A  |`  _V )  ~<_  om )  ->  dom  ( A  |`  _V )  ~<_  om )
2317, 21, 22syl2anc 673 . 2  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  om )
241, 23syl5eqbrr 4430 1  |-  ( A  ~<_  om  ->  dom  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   Rel wrel 4844    Fn wfn 5584   -onto->wfo 5587   ` cfv 5589   omcom 6711   1stc1st 6810    ~<_ cdom 7585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-ac2 8911
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-card 8391  df-acn 8394  df-ac 8565
This theorem is referenced by:  rnct  28375
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