MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numwdom Unicode version

Theorem numwdom 7686
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 7282 . 2  |-  ( B  ~<_*  A  ->  ( B  =  (/)  \/  E. f  f : A -onto-> B ) )
2 simpr 447 . . . 4  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  =  (/) )
3 0fin 7087 . . . . 5  |-  (/)  e.  Fin
4 finnum 7581 . . . . 5  |-  ( (/)  e.  Fin  ->  (/)  e.  dom  card )
53, 4ax-mp 8 . . . 4  |-  (/)  e.  dom  card
62, 5syl6eqel 2371 . . 3  |-  ( ( A  e.  dom  card  /\  B  =  (/) )  ->  B  e.  dom  card )
7 fonum 7685 . . . . . 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 1664 . . . 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 1528    = wceq 1623    e. wcel 1684   (/)c0 3455   class class class wbr 4023   dom cdm 4689   -onto->wfo 5253   Fincfn 6863    ~<_* cwdom 7271   cardccrd 7568
This theorem is referenced by:  ptcmplem2  17747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-fin 6867  df-wdom 7273  df-card 7572  df-acn 7575
  Copyright terms: Public domain W3C validator