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Theorem wdomnumr 8462
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 8011 . . 3  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  <->  ( A  =  (/)  \/  E. x  x : B -onto-> A ) ) )
2 0domg 7663 . . . . 5  |-  ( B  e.  dom  card  ->  (/)  ~<_  B )
3 breq1 4459 . . . . 5  |-  ( A  =  (/)  ->  ( A  ~<_  B  <->  (/)  ~<_  B ) )
42, 3syl5ibrcom 222 . . . 4  |-  ( B  e.  dom  card  ->  ( A  =  (/)  ->  A  ~<_  B ) )
5 fodomnum 8455 . . . . 5  |-  ( B  e.  dom  card  ->  ( x : B -onto-> A  ->  A  ~<_  B ) )
65exlimdv 1725 . . . 4  |-  ( B  e.  dom  card  ->  ( E. x  x : B -onto-> A  ->  A  ~<_  B ) )
74, 6jaod 380 . . 3  |-  ( B  e.  dom  card  ->  ( ( A  =  (/)  \/ 
E. x  x : B -onto-> A )  ->  A  ~<_  B ) )
81, 7sylbid 215 . 2  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  ->  A  ~<_  B ) )
9 domwdom 8018 . 2  |-  ( A  ~<_  B  ->  A  ~<_*  B )
108, 9impbid1 203 1  |-  ( B  e.  dom  card  ->  ( A  ~<_*  B  <->  A  ~<_  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1395   E.wex 1613    e. wcel 1819   (/)c0 3793   class class class wbr 4456   dom cdm 5008   -onto->wfo 5592    ~<_ cdom 7533    ~<_* cwdom 8001   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-recs 7060  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-wdom 8003  df-card 8337  df-acn 8340
This theorem is referenced by:  wdomac  8922  ttac  31182  isnumbasgrplem2  31257
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