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Theorem numwdom 7833
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 7429 . 2  |-  ( B  ~<_*  A  ->  ( B  =  (/)  \/  E. f  f : A -onto-> B ) )
2 simpr 447 . . . 4  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  =  (/) )
3 0fin 7233 . . . . 5  |-  (/)  e.  Fin
4 finnum 7728 . . . . 5  |-  ( (/)  e.  Fin  ->  (/)  e.  dom  card )
53, 4ax-mp 8 . . . 4  |-  (/)  e.  dom  card
62, 5syl6eqel 2454 . . 3  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  e.  dom  card )
7 fonum 7832 . . . . . 6  |-  ( ( A  e.  dom  card  /\  f : A -onto-> B
)  ->  B  e.  dom  card )
87ex 423 . . . . 5  |-  ( A  e.  dom  card  ->  ( f : A -onto-> B  ->  B  e.  dom  card ) )
98exlimdv 1641 . . . 4  |-  ( A  e.  dom  card  ->  ( E. f  f : A -onto-> B  ->  B  e. 
dom  card ) )
109imp 418 . . 3  |-  ( ( A  e.  dom  card  /\ 
E. f  f : A -onto-> B )  ->  B  e.  dom  card )
116, 10jaodan 760 . 2  |-  ( ( A  e.  dom  card  /\  ( B  =  (/)  \/ 
E. f  f : A -onto-> B ) )  ->  B  e.  dom  card )
121, 11sylan2 460 1  |-  ( ( A  e.  dom  card  /\  B  ~<_*  A )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1546    = wceq 1647    e. wcel 1715   (/)c0 3543   class class class wbr 4125   dom cdm 4792   -onto->wfo 5356   Fincfn 7006    ~<_* cwdom 7418   cardccrd 7715
This theorem is referenced by:  ptcmplem2  17960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-fin 7010  df-wdom 7420  df-card 7719  df-acn 7722
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