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Theorem xpdom1g 7511
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 7419 . . . 4  |-  Rel  ~<_
21brrelexi 4980 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
3 xpcomeng 7506 . . . 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 7510 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
71brrelex2i 4981 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
8 xpcomeng 7506 . . . 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 7471 . . 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 7470 . 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 1758   _Vcvv 3071   class class class wbr 4393    X. cxp 4939    ~~ cen 7410    ~<_ cdom 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-1st 6680  df-2nd 6681  df-en 7414  df-dom 7415
This theorem is referenced by:  xpdom1  7513  xpen  7577  infpwfien  8336  iunctb  8842  canthp1lem1  8923  gchxpidm  8940  xpct  26154  fnct  26157
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