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Theorem xpdom1g 7611
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
StepHypRef Expression
1 reldom 7519 . . . 4  |-  Rel  ~<_
21brrelexi 5039 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
3 xpcomeng 7606 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  V )  ->  ( A  X.  C
)  ~~  ( C  X.  A ) )
43ancoms 453 . . 3  |-  ( ( C  e.  V  /\  A  e.  _V )  ->  ( A  X.  C
)  ~~  ( C  X.  A ) )
52, 4sylan2 474 . 2  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~~  ( C  X.  A ) )
6 xpdom2g 7610 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
71brrelex2i 5040 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
8 xpcomeng 7606 . . . 4  |-  ( ( C  e.  V  /\  B  e.  _V )  ->  ( C  X.  B
)  ~~  ( B  X.  C ) )
97, 8sylan2 474 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  B
)  ~~  ( B  X.  C ) )
10 domentr 7571 . . 3  |-  ( ( ( C  X.  A
)  ~<_  ( C  X.  B )  /\  ( C  X.  B )  ~~  ( B  X.  C
) )  ->  ( C  X.  A )  ~<_  ( B  X.  C ) )
116, 9, 10syl2anc 661 . 2  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( B  X.  C ) )
12 endomtr 7570 . 2  |-  ( ( ( A  X.  C
)  ~~  ( C  X.  A )  /\  ( C  X.  A )  ~<_  ( B  X.  C ) )  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
135, 11, 12syl2anc 661 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   _Vcvv 3113   class class class wbr 4447    X. cxp 4997    ~~ cen 7510    ~<_ cdom 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1st 6781  df-2nd 6782  df-en 7514  df-dom 7515
This theorem is referenced by:  xpdom1  7613  xpen  7677  infpwfien  8439  iunctb  8945  canthp1lem1  9026  gchxpidm  9043  xpct  27205  fnct  27208
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