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Theorem numwdom 7904
Description: A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
numwdom  |-  ( ( A  e.  dom  card  /\  B  ~<_*  A )  ->  B  e.  dom  card )

Proof of Theorem numwdom
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brwdomi 7500 . 2  |-  ( B  ~<_*  A  ->  ( B  =  (/)  \/  E. f  f : A -onto-> B ) )
2 simpr 448 . . . 4  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  =  (/) )
3 0fin 7303 . . . . 5  |-  (/)  e.  Fin
4 finnum 7799 . . . . 5  |-  ( (/)  e.  Fin  ->  (/)  e.  dom  card )
53, 4ax-mp 8 . . . 4  |-  (/)  e.  dom  card
62, 5syl6eqel 2500 . . 3  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  e.  dom  card )
7 fonum 7903 . . . . . 6  |-  ( ( A  e.  dom  card  /\  f : A -onto-> B
)  ->  B  e.  dom  card )
87ex 424 . . . . 5  |-  ( A  e.  dom  card  ->  ( f : A -onto-> B  ->  B  e.  dom  card ) )
98exlimdv 1643 . . . 4  |-  ( A  e.  dom  card  ->  ( E. f  f : A -onto-> B  ->  B  e. 
dom  card ) )
109imp 419 . . 3  |-  ( ( A  e.  dom  card  /\ 
E. f  f : A -onto-> B )  ->  B  e.  dom  card )
116, 10jaodan 761 . 2  |-  ( ( A  e.  dom  card  /\  ( B  =  (/)  \/ 
E. f  f : A -onto-> B ) )  ->  B  e.  dom  card )
121, 11sylan2 461 1  |-  ( ( A  e.  dom  card  /\  B  ~<_*  A )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   (/)c0 3596   class class class wbr 4180   dom cdm 4845   -onto->wfo 5419   Fincfn 7076    ~<_* cwdom 7489   cardccrd 7786
This theorem is referenced by:  ptcmplem2  18045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-fin 7080  df-wdom 7491  df-card 7790  df-acn 7793
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