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Theorem numwdom 7945
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 7539 . 2  |-  ( B  ~<_*  A  ->  ( B  =  (/)  \/  E. f  f : A -onto-> B ) )
2 simpr 449 . . . 4  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  =  (/) )
3 0fin 7339 . . . . 5  |-  (/)  e.  Fin
4 finnum 7840 . . . . 5  |-  ( (/)  e.  Fin  ->  (/)  e.  dom  card )
53, 4ax-mp 5 . . . 4  |-  (/)  e.  dom  card
62, 5syl6eqel 2526 . . 3  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  e.  dom  card )
7 fonum 7944 . . . . . 6  |-  ( ( A  e.  dom  card  /\  f : A -onto-> B
)  ->  B  e.  dom  card )
87ex 425 . . . . 5  |-  ( A  e.  dom  card  ->  ( f : A -onto-> B  ->  B  e.  dom  card ) )
98exlimdv 1647 . . . 4  |-  ( A  e.  dom  card  ->  ( E. f  f : A -onto-> B  ->  B  e. 
dom  card ) )
109imp 420 . . 3  |-  ( ( A  e.  dom  card  /\ 
E. f  f : A -onto-> B )  ->  B  e.  dom  card )
116, 10jaodan 762 . 2  |-  ( ( A  e.  dom  card  /\  ( B  =  (/)  \/ 
E. f  f : A -onto-> B ) )  ->  B  e.  dom  card )
121, 11sylan2 462 1  |-  ( ( A  e.  dom  card  /\  B  ~<_*  A )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   (/)c0 3630   class class class wbr 4215   dom cdm 4881   -onto->wfo 5455   Fincfn 7112    ~<_* cwdom 7528   cardccrd 7827
This theorem is referenced by:  ptcmplem2  18089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-fin 7116  df-wdom 7530  df-card 7831  df-acn 7834
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