Theorem wdomen2 8097

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 7601	. . . 4 |- ( A ~~ B -> A ~<_ B )
3		domwdom 8094	. . . 4 |- ( A ~<_ B -> A ~<_* B )
4	2, 3	syl 17	. . 3 |- ( A ~~ B -> A ~<_* B )
5		wdomtr 8095	. . 3 |- ( ( C ~<_* A /\ A ~<_* B ) -> C ~<_* B )
6	1, 4, 5	syl2anr 481	. 2 |- ( ( A ~~ B /\ C ~<_* A ) -> C ~<_* B )
7		id 22	. . 3 |- ( C ~<_* B -> C ~<_* B )
8		ensym 7623	. . . 4 |- ( A ~~ B -> B ~~ A )
9		endom 7601	. . . 4 |- ( B ~~ A -> B ~<_ A )
10		domwdom 8094	. . . 4 |- ( B ~<_ A -> B ~<_* A )
11	8, 9, 10	3syl 18	. . 3 |- ( A ~~ B -> B ~<_* A )
12		wdomtr 8095	. . 3 |- ( ( C ~<_* B /\ B ~<_* A ) -> C ~<_* A )
13	7, 11, 12	syl2anr 481	. 2 |- ( ( A ~~ B /\ C ~<_* B ) -> C ~<_* A )
14	6, 13	impbida 844	1 |- ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) )

Syntax hints:   -> wi 4   <-> wb 188   class class class wbr 4405   ~~ cen 7571   ~<_ cdom 7572   ~<_^* cwdom 8077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-wdom 8079

This theorem is referenced by: (None)