Theorem wdomen2 7994

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 7534	. . . 4 |- ( A ~~ B -> A ~<_ B )
3		domwdom 7991	. . . 4 |- ( A ~<_ B -> A ~<_* B )
4	2, 3	syl 16	. . 3 |- ( A ~~ B -> A ~<_* B )
5		wdomtr 7992	. . 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 7556	. . . 4 |- ( A ~~ B -> B ~~ A )
9		endom 7534	. . . 4 |- ( B ~~ A -> B ~<_ A )
10		domwdom 7991	. . . 4 |- ( B ~<_ A -> B ~<_* A )
11	8, 9, 10	3syl 20	. . 3 |- ( A ~~ B -> B ~<_* A )
12		wdomtr 7992	. . 3 |- ( ( C ~<_* B /\ B ~<_* A ) -> C ~<_* A )
13	7, 11, 12	syl2anr 478	. 2 |- ( ( A ~~ B /\ C ~<_* B ) -> C ~<_* A )
14	6, 13	impbida 829	1 |- ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) )

Syntax hints:   -> wi 4   <-> wb 184   class class class wbr 4442   ~~ cen 7505   ~<_ cdom 7506   ~<_^* cwdom 7974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-wdom 7976

This theorem is referenced by: (None)