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Theorem ssxpb 5351
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 5336 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 dmxp 5134 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32adantl 464 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  =  A )
41, 3sylbir 213 . . . . . . 7  |-  ( ( A  X.  B )  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
54adantr 463 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  dom  ( A  X.  B
)  =  A )
6 dmss 5115 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 464 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
85, 7eqsstr3d 3452 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_ 
dom  ( C  X.  D ) )
9 dmxpss 5348 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9syl6ss 3429 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_  C )
11 rnxp 5347 . . . . . . . . 9  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
1211adantr 463 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  ran  ( A  X.  B
)  =  B )
131, 12sylbir 213 . . . . . . 7  |-  ( ( A  X.  B )  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
1413adantr 463 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ran  ( A  X.  B
)  =  B )
15 rnss 5144 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 464 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1714, 16eqsstr3d 3452 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_ 
ran  ( C  X.  D ) )
18 rnxpss 5349 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18syl6ss 3429 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_  D )
2010, 19jca 530 . . 3  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ( A  C_  C  /\  B  C_  D ) )
2120ex 432 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
22 xpss12 5021 . 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 367    = wceq 1399    =/= wne 2577    C_ wss 3389   (/)c0 3711    X. cxp 4911   dom cdm 4913   ran crn 4914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924
This theorem is referenced by:  xp11  5352  dibord  37299
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