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Theorem xpdom1g 7633
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 7541 . . . 4  |-  Rel  ~<_
21brrelexi 5049 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
3 xpcomeng 7628 . . . 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 7632 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
71brrelex2i 5050 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
8 xpcomeng 7628 . . . 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 7593 . . 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 7592 . 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 1819   _Vcvv 3109   class class class wbr 4456    X. cxp 5006    ~~ cen 7532    ~<_ cdom 7533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1st 6799  df-2nd 6800  df-en 7536  df-dom 7537
This theorem is referenced by:  xpdom1  7635  xpen  7699  infpwfien  8460  iunctb  8966  canthp1lem1  9047  gchxpidm  9064  xpct  27683  fnct  27686
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