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

Step	Hyp	Ref	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 )
4	2, 3	syl 16	. . 3 |- ( A ~~ B -> A ~<_* B )
5		wdomtr 7789	. . 3 |- ( ( C ~<_* A /\ A ~<_* B ) -> C ~<_* B )
6	1, 4, 5	syl2anr 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 )
11	8, 9, 10	3syl 20	. . 3 |- ( A ~~ B -> B ~<_* A )
12		wdomtr 7789	. . 3 |- ( ( C ~<_* B /\ B ~<_* A ) -> C ~<_* A )
13	7, 11, 12	syl2anr 478	. 2 |- ( ( A ~~ B /\ C ~<_* B ) -> C ~<_* A )
14	6, 13	impbida 828	1 |- ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) )

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)