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Theorem xpdom1g 6503
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 6416 . . . 4  |-  Rel  ~<_
21brrelexi 4452 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
3 xpcomeng 6498 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  V )  ->  ( A  X.  C
)  ~~  ( C  X.  A ) )
43ancoms 264 . . 3  |-  ( ( C  e.  V  /\  A  e.  _V )  ->  ( A  X.  C
)  ~~  ( C  X.  A ) )
52, 4sylan2 280 . 2  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~~  ( C  X.  A ) )
6 xpdom2g 6502 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
71brrelex2i 4453 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
8 xpcomeng 6498 . . . 4  |-  ( ( C  e.  V  /\  B  e.  _V )  ->  ( C  X.  B
)  ~~  ( B  X.  C ) )
97, 8sylan2 280 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  B
)  ~~  ( B  X.  C ) )
10 domentr 6462 . . 3  |-  ( ( ( C  X.  A
)  ~<_  ( C  X.  B )  /\  ( C  X.  B )  ~~  ( B  X.  C
) )  ->  ( C  X.  A )  ~<_  ( B  X.  C ) )
116, 9, 10syl2anc 403 . 2  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( B  X.  C ) )
12 endomtr 6461 . 2  |-  ( ( ( A  X.  C
)  ~~  ( C  X.  A )  /\  ( C  X.  A )  ~<_  ( B  X.  C ) )  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
135, 11, 12syl2anc 403 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1436   _Vcvv 2615   class class class wbr 3822    X. cxp 4411    ~~ cen 6409    ~<_ cdom 6410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-1st 5870  df-2nd 5871  df-en 6412  df-dom 6413
This theorem is referenced by:  xpdom1  6505  xpct  11134
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