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Theorem wdomen2 7791
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
wdomen2  |-  ( A 
~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )

Proof of Theorem wdomen2
StepHypRef Expression
1 id 22 . . 3  |-  ( C  ~<_*  A  ->  C  ~<_*  A )
2 endom 7335 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
3 domwdom 7788 . . . 4  |-  ( A  ~<_  B  ->  A  ~<_*  B )
42, 3syl 16 . . 3  |-  ( A 
~~  B  ->  A  ~<_*  B )
5 wdomtr 7789 . . 3  |-  ( ( C  ~<_*  A  /\  A  ~<_*  B
)  ->  C  ~<_*  B )
61, 4, 5syl2anr 478 . 2  |-  ( ( A  ~~  B  /\  C  ~<_*  A )  ->  C  ~<_*  B )
7 id 22 . . 3  |-  ( C  ~<_*  B  ->  C  ~<_*  B )
8 ensym 7357 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
9 endom 7335 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
10 domwdom 7788 . . . 4  |-  ( B  ~<_  A  ->  B  ~<_*  A )
118, 9, 103syl 20 . . 3  |-  ( A 
~~  B  ->  B  ~<_*  A )
12 wdomtr 7789 . . 3  |-  ( ( C  ~<_*  B  /\  B  ~<_*  A
)  ->  C  ~<_*  A )
137, 11, 12syl2anr 478 . 2  |-  ( ( A  ~~  B  /\  C  ~<_*  B )  ->  C  ~<_*  A )
146, 13impbida 828 1  |-  ( A 
~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4291    ~~ cen 7306    ~<_ cdom 7307    ~<_* cwdom 7771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-wdom 7773
This theorem is referenced by: (None)
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