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Theorem numdom 7853
Description: A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
numdom  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )

Proof of Theorem numdom
StepHypRef Expression
1 cardon 7765 . 2  |-  ( card `  A )  e.  On
2 cardid2 7774 . . . 4  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
3 domen2 7187 . . . 4  |-  ( (
card `  A )  ~~  A  ->  ( B  ~<_  ( card `  A
)  <->  B  ~<_  A )
)
42, 3syl 16 . . 3  |-  ( A  e.  dom  card  ->  ( B  ~<_  ( card `  A
)  <->  B  ~<_  A )
)
54biimpar 472 . 2  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  ~<_  ( card `  A
) )
6 ondomen 7852 . 2  |-  ( ( ( card `  A
)  e.  On  /\  B  ~<_  ( card `  A
) )  ->  B  e.  dom  card )
71, 5, 6sylancr 645 1  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   class class class wbr 4154   Oncon0 4523   dom cdm 4819   ` cfv 5395    ~~ cen 7043    ~<_ cdom 7044   cardccrd 7756
This theorem is referenced by:  ssnum  7854  indcardi  7856  fonum  7873  infpwfien  7877  inffien  7878  unnum  8014  infdif  8023  infxpabs  8026  infunsdom1  8027  infunsdom  8028  infmap2  8032  gchac  8482  grothac  8639  mbfimaopnlem  19415  ttac  26799  isnumbasgrplem2  26939
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-riota 6486  df-recs 6570  df-er 6842  df-en 7047  df-dom 7048  df-card 7760
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