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Theorem numdom 8410
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 8316 . 2  |-  ( card `  A )  e.  On
2 cardid2 8325 . . . 4  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
3 domen2 7652 . . . 4  |-  ( (
card `  A )  ~~  A  ->  ( B  ~<_  ( card `  A
)  <->  B  ~<_  A )
)
42, 3syl 16 . . 3  |-  ( A  e.  dom  card  ->  ( B  ~<_  ( card `  A
)  <->  B  ~<_  A )
)
54biimpar 485 . 2  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  ~<_  ( card `  A
) )
6 ondomen 8409 . 2  |-  ( ( ( card `  A
)  e.  On  /\  B  ~<_  ( card `  A
) )  ->  B  e.  dom  card )
71, 5, 6sylancr 663 1  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762   class class class wbr 4442   Oncon0 4873   dom cdm 4994   ` cfv 5581    ~~ cen 7505    ~<_ cdom 7506   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-recs 7034  df-er 7303  df-en 7509  df-dom 7510  df-card 8311
This theorem is referenced by:  ssnum  8411  indcardi  8413  fonum  8430  infpwfien  8434  inffien  8435  unnum  8571  infdif  8580  infxpabs  8583  infunsdom1  8584  infunsdom  8585  infmap2  8589  gchac  9050  grothac  9199  mbfimaopnlem  21792  ttac  30573  isnumbasgrplem2  30648
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