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Theorem dmct 27969
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 5281 . 2  |-  dom  ( A  |`  _V )  =  dom  A
2 resss 5116 . . . . 5  |-  ( A  |`  _V )  C_  A
3 ctex 27963 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
4 ssexg 4539 . . . . 5  |-  ( ( ( A  |`  _V )  C_  A  /\  A  e. 
_V )  ->  ( A  |`  _V )  e. 
_V )
52, 3, 4sylancr 661 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  e.  _V )
6 fvex 5858 . . . . . . 7  |-  ( 1st `  x )  e.  _V
7 eqid 2402 . . . . . . 7  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  =  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x
) )
86, 7fnmpti 5691 . . . . . 6  |-  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) )  Fn  ( A  |`  _V )
9 dffn4 5783 . . . . . 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 5120 . . . . . 6  |-  Rel  ( A  |`  _V )
12 reldm 6834 . . . . . 6  |-  ( Rel  ( A  |`  _V )  ->  dom  ( A  |`  _V )  =  ran  ( x  e.  ( A  |`  _V )  |->  ( 1st `  x ) ) )
13 foeq3 5775 . . . . . 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 8934 . . . 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 19 . . 3  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  ( A  |`  _V ) )
18 ssdomg 7598 . . . . 5  |-  ( A  e.  _V  ->  (
( A  |`  _V )  C_  A  ->  ( A  |` 
_V )  ~<_  A ) )
193, 2, 18mpisyl 19 . . . 4  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  A )
20 domtr 7605 . . . 4  |-  ( ( ( A  |`  _V )  ~<_  A  /\  A  ~<_  om )  ->  ( A  |`  _V )  ~<_  om )
2119, 20mpancom 667 . . 3  |-  ( A  ~<_  om  ->  ( A  |` 
_V )  ~<_  om )
22 domtr 7605 . . 3  |-  ( ( dom  ( A  |`  _V )  ~<_  ( A  |` 
_V )  /\  ( A  |`  _V )  ~<_  om )  ->  dom  ( A  |`  _V )  ~<_  om )
2317, 21, 22syl2anc 659 . 2  |-  ( A  ~<_  om  ->  dom  ( A  |`  _V )  ~<_  om )
241, 23syl5eqbrr 4428 1  |-  ( A  ~<_  om  ->  dom  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    |-> cmpt 4452   dom cdm 4822   ran crn 4823    |` cres 4824   Rel wrel 4827    Fn wfn 5563   -onto->wfo 5566   ` cfv 5568   omcom 6682   1stc1st 6781    ~<_ cdom 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-ac2 8874
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-card 8351  df-acn 8354  df-ac 8528
This theorem is referenced by:  rnct  27971
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