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Theorem dmct 27209
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 5463 . 2  |-  dom  ( A  |`  _V )  =  dom  A
2 resss 5295 . . . . 5  |-  ( A  |`  _V )  C_  A
3 ctex 27203 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
4 ssexg 4593 . . . . 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 5874 . . . . . . 7  |-  ( 1st `  x )  e.  _V
7 eqid 2467 . . . . . . 7  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
86, 7fnmpti 5707 . . . . . 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 208 . . . . 5  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
11 relres 5299 . . . . . 6  |-  Rel  ( A  |`  _V )
12 reldm 6832 . . . . . 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 209 . . . 4  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) : ( A  |`  _V ) -onto-> dom  ( A  |`  _V )
16 fodomg 8899 . . . 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 7558 . . . . 5  |-  ( A  e.  _V  ->  (
( A  |`  _V )  C_  A  ->  ( A  |` 
_V )  ~<_  A ) )
193, 2, 18mpisyl 18 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  A )
20 domtr 7565 . . . 4  |-  ( ( ( A  |`  _V )  ~<_  A  /\  A  ~<_  om )  ->  ( A  |`  _V )  ~<_  om )
2119, 20mpancom 669 . . 3  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  om )
22 domtr 7565 . . 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 4481 1  |-  ( A  ~<_  om  ->  dom  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000    |` cres 5001   Rel wrel 5004    Fn wfn 5581   -onto->wfo 5584   ` cfv 5586   omcom 6678   1stc1st 6779    ~<_ cdom 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-ac2 8839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-recs 7039  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-card 8316  df-acn 8319  df-ac 8493
This theorem is referenced by:  rnct  27211
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