Theorem xpdom1g 7511

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 7419	. . . 4 |- Rel ~<_
2	1	brrelexi 4980	. . 3 |- ( A ~<_ B -> A e. _V )
3		xpcomeng 7506	. . . 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 7510	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
7	1	brrelex2i 4981	. . . 4 |- ( A ~<_ B -> B e. _V )
8		xpcomeng 7506	. . . 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 7471	. . 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 7470	. 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 1758   _Vcvv 3071   class class class wbr 4393   X. cxp 4939   ~~ cen 7410   ~<_ cdom 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-1st 6680  df-2nd 6681  df-en 7414  df-dom 7415

This theorem is referenced by:  xpdom1  7513  xpen  7577  infpwfien  8336  iunctb  8842  canthp1lem1  8923  gchxpidm  8940  xpct  26154  fnct  26157