Theorem xpdom1g 7611

Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
xpdom1g	|- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )

Proof of Theorem xpdom1g

Step	Hyp	Ref	Expression
1		reldom 7519	. . . 4 |- Rel ~<_
2	1	brrelexi 5039	. . 3 |- ( A ~<_ B -> A e. _V )
3		xpcomeng 7606	. . . 4 |- ( ( A e. _V /\ C e. V ) -> ( A X. C ) ~~ ( C X. A ) )
4	3	ancoms 453	. . 3 |- ( ( C e. V /\ A e. _V ) -> ( A X. C ) ~~ ( C X. A ) )
5	2, 4	sylan2 474	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~~ ( C X. A ) )
6		xpdom2g 7610	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
7	1	brrelex2i 5040	. . . 4 |- ( A ~<_ B -> B e. _V )
8		xpcomeng 7606	. . . 4 |- ( ( C e. V /\ B e. _V ) -> ( C X. B ) ~~ ( B X. C ) )
9	7, 8	sylan2 474	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. B ) ~~ ( B X. C ) )
10		domentr 7571	. . 3 |- ( ( ( C X. A ) ~<_ ( C X. B ) /\ ( C X. B ) ~~ ( B X. C ) ) -> ( C X. A ) ~<_ ( B X. C ) )
11	6, 9, 10	syl2anc 661	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( B X. C ) )
12		endomtr 7570	. 2 |- ( ( ( A X. C ) ~~ ( C X. A ) /\ ( C X. A ) ~<_ ( B X. C ) ) -> ( A X. C ) ~<_ ( B X. C ) )
13	5, 11, 12	syl2anc 661	1 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1767   _Vcvv 3113   class class class wbr 4447   X. cxp 4997   ~~ cen 7510   ~<_ cdom 7511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1st 6781  df-2nd 6782  df-en 7514  df-dom 7515

This theorem is referenced by:  xpdom1  7613  xpen  7677  infpwfien  8439  iunctb  8945  canthp1lem1  9026  gchxpidm  9043  xpct  27205  fnct  27208