Theorem xpdom1g 6811

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 6723	. . . 4 |- Rel ~<_
2	1	brrelex1i 4654	. . 3 |- ( A ~<_ B -> A e. _V )
3		xpcomeng 6806	. . . 4 |- ( ( A e. _V /\ C e. V ) -> ( A X. C ) ~~ ( C X. A ) )
4	3	ancoms 266	. . 3 |- ( ( C e. V /\ A e. _V ) -> ( A X. C ) ~~ ( C X. A ) )
5	2, 4	sylan2 284	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~~ ( C X. A ) )
6		xpdom2g 6810	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
7	1	brrelex2i 4655	. . . 4 |- ( A ~<_ B -> B e. _V )
8		xpcomeng 6806	. . . 4 |- ( ( C e. V /\ B e. _V ) -> ( C X. B ) ~~ ( B X. C ) )
9	7, 8	sylan2 284	. . 3 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. B ) ~~ ( B X. C ) )
10		domentr 6769	. . 3 |- ( ( ( C X. A ) ~<_ ( C X. B ) /\ ( C X. B ) ~~ ( B X. C ) ) -> ( C X. A ) ~<_ ( B X. C ) )
11	6, 9, 10	syl2anc 409	. 2 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( B X. C ) )
12		endomtr 6768	. 2 |- ( ( ( A X. C ) ~~ ( C X. A ) /\ ( C X. A ) ~<_ ( B X. C ) ) -> ( A X. C ) ~<_ ( B X. C ) )
13	5, 11, 12	syl2anc 409	1 |- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   X. cxp 4609   ~~ cen 6716   ~<_ cdom 6717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-en 6719  df-dom 6720

This theorem is referenced by:  xpdom1  6813  xpct  12351