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Theorem ssxpb 5427
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 5412 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 dmxp 5207 . . . . . . . . 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 5188 . . . . . . 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 3521 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_ 
dom  ( C  X.  D ) )
9 dmxpss 5424 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9syl6ss 3498 . . . 4  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  A  C_  C )
11 rnxp 5423 . . . . . . . . 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 5217 . . . . . . 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 3521 . . . . 5  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  C_  ( C  X.  D
) )  ->  B  C_ 
ran  ( C  X.  D ) )
18 rnxpss 5425 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18syl6ss 3498 . . . 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 5094 . 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 1381    =/= wne 2636    C_ wss 3458   (/)c0 3767    X. cxp 4983   dom cdm 4985   ran crn 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-cnv 4993  df-dm 4995  df-rn 4996
This theorem is referenced by:  xp11  5428  dibord  36588
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