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Theorem ssxpb 5375
Description: A Cartesian product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
ssxpb  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )

Proof of Theorem ssxpb
StepHypRef Expression
1 xpnz 5360 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 dmxp 5161 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32adantl 466 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  =  A )
41, 3sylbir 213 . . . . . . 7  |-  ( ( A  X.  B )  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
54adantr 465 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  dom  ( A  X.  B
)  =  A )
6 dmss 5142 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 466 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
85, 7eqsstr3d 3494 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_ 
dom  ( C  X.  D ) )
9 dmxpss 5372 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9syl6ss 3471 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_  C )
11 rnxp 5371 . . . . . . . . 9  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
1211adantr 465 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  ran  ( A  X.  B
)  =  B )
131, 12sylbir 213 . . . . . . 7  |-  ( ( A  X.  B )  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
1413adantr 465 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ran  ( A  X.  B
)  =  B )
15 rnss 5171 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 466 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1714, 16eqsstr3d 3494 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_ 
ran  ( C  X.  D ) )
18 rnxpss 5373 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18syl6ss 3471 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_  D )
2010, 19jca 532 . . 3  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ( A  C_  C  /\  B  C_  D ) )
2120ex 434 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
22 xpss12 5048 . 2  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( A  X.  B
)  C_  ( C  X.  D ) )
2321, 22impbid1 203 1  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    =/= wne 2645    C_ wss 3431   (/)c0 3740    X. cxp 4941   dom cdm 4943   ran crn 4944
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-cnv 4951  df-dm 4953  df-rn 4954
This theorem is referenced by:  xp11  5376  dibord  35123
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