Theorem wdomen2 8021

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 7561	. . . 4 |- ( A ~~ B -> A ~<_ B )
3		domwdom 8018	. . . 4 |- ( A ~<_ B -> A ~<_* B )
4	2, 3	syl 16	. . 3 |- ( A ~~ B -> A ~<_* B )
5		wdomtr 8019	. . 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 7583	. . . 4 |- ( A ~~ B -> B ~~ A )
9		endom 7561	. . . 4 |- ( B ~~ A -> B ~<_ A )
10		domwdom 8018	. . . 4 |- ( B ~<_ A -> B ~<_* A )
11	8, 9, 10	3syl 20	. . 3 |- ( A ~~ B -> B ~<_* A )
12		wdomtr 8019	. . 3 |- ( ( C ~<_* B /\ B ~<_* A ) -> C ~<_* A )
13	7, 11, 12	syl2anr 478	. 2 |- ( ( A ~~ B /\ C ~<_* B ) -> C ~<_* A )
14	6, 13	impbida 832	1 |- ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) )

Syntax hints:   -> wi 4   <-> wb 184   class class class wbr 4456   ~~ cen 7532   ~<_ cdom 7533   ~<_^* cwdom 8001

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-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-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-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  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-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-wdom 8003

This theorem is referenced by: (None)