Theorem wdomnumr 8887

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.

Step	Hyp	Ref	Expression
1		brwdom 8472	. . 3 |- ( B e. dom card -> ( A ~<_* B <-> ( A = (/) \/ E. x x : B -onto-> A ) ) )
2		0domg 8087	. . . . 5 |- ( B e. dom card -> (/) ~<_ B )
3		breq1 4656	. . . . 5 |- ( A = (/) -> ( A ~<_ B <-> (/) ~<_ B ) )
4	2, 3	syl5ibrcom 237	. . . 4 |- ( B e. dom card -> ( A = (/) -> A ~<_ B ) )
5		fodomnum 8880	. . . . 5 |- ( B e. dom card -> ( x : B -onto-> A -> A ~<_ B ) )
6	5	exlimdv 1861	. . . 4 |- ( B e. dom card -> ( E. x x : B -onto-> A -> A ~<_ B ) )
7	4, 6	jaod 395	. . 3 |- ( B e. dom card -> ( ( A = (/) \/ E. x x : B -onto-> A ) -> A ~<_ B ) )
8	1, 7	sylbid 230	. 2 |- ( B e. dom card -> ( A ~<_* B -> A ~<_ B ) )
9		domwdom 8479	. 2 |- ( A ~<_ B -> A ~<_* B )
10	8, 9	impbid1 215	1 |- ( B e. dom card -> ( A ~<_* B <-> A ~<_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   class class class wbr 4653   dom cdm 5114   -onto->wfo 5886   ~<_ cdom 7953   ~<_^* cwdom 8462   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464  df-card 8765  df-acn 8768

This theorem is referenced by:  wdomac  9349  ttac  37603  isnumbasgrplem2  37674