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Theorem ssxpb 5295
Description: A cross-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 5284 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 dmxp 5080 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32adantl 453 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  =  A )
41, 3sylbir 205 . . . . . . 7  |-  ( ( A  X.  B )  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
54adantr 452 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  dom  ( A  X.  B
)  =  A )
6 dmss 5061 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 453 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
85, 7eqsstr3d 3375 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_ 
dom  ( C  X.  D ) )
9 dmxpss 5292 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9syl6ss 3352 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_  C )
11 rnxp 5291 . . . . . . . . 9  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
1211adantr 452 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  ran  ( A  X.  B
)  =  B )
131, 12sylbir 205 . . . . . . 7  |-  ( ( A  X.  B )  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
1413adantr 452 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ran  ( A  X.  B
)  =  B )
15 rnss 5090 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 453 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1714, 16eqsstr3d 3375 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_ 
ran  ( C  X.  D ) )
18 rnxpss 5293 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18syl6ss 3352 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_  D )
2010, 19jca 519 . . 3  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  ( A  C_  C  /\  B  C_  D ) )
2120ex 424 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
22 xpss12 4973 . 2  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( A  X.  B
)  C_  ( C  X.  D ) )
2321, 22impbid1 195 1  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    =/= wne 2598    C_ wss 3312   (/)c0 3620    X. cxp 4868   dom cdm 4870   ran crn 4871
This theorem is referenced by:  xp11  5296  dibord  31884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881
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