Theorem xpdom1 4423

Description: Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149.

Hypotheses

Ref	Expression
xpdom.1	|- B e. V
xpdom.2	|- C e. V

Assertion

Ref	Expression
xpdom1	|- (A ~<_ B -> (A X. C) ~<_ (B X. C))

Proof of Theorem xpdom1

Step	Hyp	Ref	Expression
1		endomtr 4401	. . 3 |- (((A X. C) ~~ (C X. A) /\ (C X. A) ~<_ (C X. B)) -> (A X. C) ~<_ (C X. B))
2		reldom 4355	. . . . 5 |- Rel ~<_
3	2	brrelexi 3198	. . . 4 |- (A ~<_ B -> A e. V)
4		xpdom.2	. . . . 5 |- C e. V
5		xpcomeng 4420	. . . . 5 |- ((A e. V /\ C e. V) -> (A X. C) ~~ (C X. A))
6	4, 5	mpan2 694	. . . 4 |- (A e. V -> (A X. C) ~~ (C X. A))
7	3, 6	syl 10	. . 3 |- (A ~<_ B -> (A X. C) ~~ (C X. A))
8		xpdom.1	. . . 4 |- B e. V
9	8, 4	xpdom2 4422	. . 3 |- (A ~<_ B -> (C X. A) ~<_ (C X. B))
10	1, 7, 9	sylanc 471	. 2 |- (A ~<_ B -> (A X. C) ~<_ (C X. B))
11	4, 8	xpcomen 4419	. . 3 |- (C X. B) ~~ (B X. C)
12		domentr 4402	. . 3 |- (((A X. C) ~<_ (C X. B) /\ (C X. B) ~~ (B X. C)) -> (A X. C) ~<_ (B X. C))
13	11, 12	mpan2 694	. 2 |- ((A X. C) ~<_ (C X. B) -> (A X. C) ~<_ (B X. C))
14	10, 13	syl 10	1 |- (A ~<_ B -> (A X. C) ~<_ (B X. C))

Syntax hints:   -> wi 3   e. wcel 955  Vcvv 1802   class class class wbr 2609   X. cxp 3158   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  xpdom1g 4424  uniimadom 4782

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-en 4351  df-dom 4352

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