Theorem xpdom1g 7121

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 7017	. . . 4 |- Rel ~<_
2	1	brrelex1i 4813	. . 3 |- ( A ~<_ B -> A e. _V )
3		xpcomeng 7116	. . . 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 7120	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
7	1	brrelex2i 4814	. . . 4 |- ( A ~<_ B -> B e. _V )
8		xpcomeng 7116	. . . 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 7068	. . 3 |- ( ( ( C X. A ) ~<_ ( C X. B ) /\ ( C X. B ) ~~ ( B X. C ) ) -> ( C X. A ) ~<_ ( B X. C ) )
11	6, 9, 10	syl2anc 415	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( B X. C ) )
12		endomtr 7067	. 2 |- ( ( ( A X. C ) ~~ ( C X. A ) /\ ( C X. A ) ~<_ ( B X. C ) ) -> ( A X. C ) ~<_ ( B X. C ) )
13	5, 11, 12	syl2anc 415	1 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   X. cxp 4767   ~~ cen 7010   ~<_ cdom 7011

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-en 7013  df-dom 7014

This theorem is referenced by:  xpdom1  7123  xpct  13265