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Theorem wdomnumr 8521
Description: Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
wdomnumr  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  <->  A  ~<_  B )
)

Proof of Theorem wdomnumr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 brwdom 8108 . . 3  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  <->  ( A  =  (/)  \/  E. x  x : B -onto-> A ) ) )
2 0domg 7725 . . . . 5  |-  ( B  e.  dom  card  ->  (/)  ~<_  B )
3 breq1 4419 . . . . 5  |-  ( A  =  (/)  ->  ( A  ~<_  B  <->  (/)  ~<_  B ) )
42, 3syl5ibrcom 230 . . . 4  |-  ( B  e.  dom  card  ->  ( A  =  (/)  ->  A  ~<_  B ) )
5 fodomnum 8514 . . . . 5  |-  ( B  e.  dom  card  ->  ( x : B -onto-> A  ->  A  ~<_  B ) )
65exlimdv 1790 . . . 4  |-  ( B  e.  dom  card  ->  ( E. x  x : B -onto-> A  ->  A  ~<_  B ) )
74, 6jaod 386 . . 3  |-  ( B  e.  dom  card  ->  ( ( A  =  (/)  \/ 
E. x  x : B -onto-> A )  ->  A  ~<_  B ) )
81, 7sylbid 223 . 2  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  ->  A  ~<_  B ) )
9 domwdom 8115 . 2  |-  ( A  ~<_  B  ->  A  ~<_*  B )
108, 9impbid1 208 1  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  <->  A  ~<_  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    = wceq 1455   E.wex 1674    e. wcel 1898   (/)c0 3743   class class class wbr 4416   dom cdm 4853   -onto->wfo 5599    ~<_ cdom 7593    ~<_* cwdom 8098   cardccrd 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-wdom 8100  df-card 8399  df-acn 8402
This theorem is referenced by:  wdomac  8981  ttac  35936  isnumbasgrplem2  36008
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