Theorem xpdom1g 6889

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 6801	. . . 4 |- Rel ~<_
2	1	brrelex1i 4703	. . 3 |- ( A ~<_ B -> A e. _V )
3		xpcomeng 6884	. . . 4 |- ( ( A e. _V /\ C e. V ) -> ( A X. C ) ~~ ( C X. A ) )
4	3	ancoms 268	. . 3 |- ( ( C e. V /\ A e. _V ) -> ( A X. C ) ~~ ( C X. A ) )
5	2, 4	sylan2 286	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~~ ( C X. A ) )
6		xpdom2g 6888	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
7	1	brrelex2i 4704	. . . 4 |- ( A ~<_ B -> B e. _V )
8		xpcomeng 6884	. . . 4 |- ( ( C e. V /\ B e. _V ) -> ( C X. B ) ~~ ( B X. C ) )
9	7, 8	sylan2 286	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. B ) ~~ ( B X. C ) )
10		domentr 6847	. . 3 |- ( ( ( C X. A ) ~<_ ( C X. B ) /\ ( C X. B ) ~~ ( B X. C ) ) -> ( C X. A ) ~<_ ( B X. C ) )
11	6, 9, 10	syl2anc 411	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( B X. C ) )
12		endomtr 6846	. 2 |- ( ( ( A X. C ) ~~ ( C X. A ) /\ ( C X. A ) ~<_ ( B X. C ) ) -> ( A X. C ) ~<_ ( B X. C ) )
13	5, 11, 12	syl2anc 411	1 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   X. cxp 4658   ~~ cen 6794   ~<_ cdom 6795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-en 6797  df-dom 6798

This theorem is referenced by:  xpdom1  6891  xpct  12556