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Theorem ssxpbm 5023
Description: A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
ssxpbm  |-  ( E. x  x  e.  ( A  X.  B )  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    C( x)    D( x)

Proof of Theorem ssxpbm
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpm 5009 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. x  x  e.  ( A  X.  B
) )
2 dmxpm 4808 . . . . . . . . 9  |-  ( E. b  b  e.  B  ->  dom  ( A  X.  B )  =  A )
32adantl 275 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  ->  dom  ( A  X.  B )  =  A )
41, 3sylbir 134 . . . . . . 7  |-  ( E. x  x  e.  ( A  X.  B )  ->  dom  ( A  X.  B )  =  A )
54adantr 274 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  dom  ( A  X.  B )  =  A )
6 dmss 4787 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 275 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  dom  ( A  X.  B )  C_  dom  ( C  X.  D
) )
85, 7eqsstrrd 3165 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  dom  ( C  X.  D
) )
9 dmxpss 5018 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9sstrdi 3140 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  C
)
11 rnxpm 5017 . . . . . . . . 9  |-  ( E. a  a  e.  A  ->  ran  ( A  X.  B )  =  B )
1211adantr 274 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  ->  ran  ( A  X.  B )  =  B )
131, 12sylbir 134 . . . . . . 7  |-  ( E. x  x  e.  ( A  X.  B )  ->  ran  ( A  X.  B )  =  B )
1413adantr 274 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ran  ( A  X.  B )  =  B )
15 rnss 4818 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 275 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ran  ( A  X.  B )  C_  ran  ( C  X.  D
) )
1714, 16eqsstrrd 3165 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  ran  ( C  X.  D
) )
18 rnxpss 5019 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18sstrdi 3140 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  D
)
2010, 19jca 304 . . 3  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ( A  C_  C  /\  B  C_  D
) )
2120ex 114 . 2  |-  ( E. x  x  e.  ( A  X.  B )  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
22 xpss12 4695 . 2  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( A  X.  B
)  C_  ( C  X.  D ) )
2321, 22impbid1 141 1  |-  ( E. x  x  e.  ( 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    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128    C_ wss 3102    X. cxp 4586   dom cdm 4588   ran crn 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599
This theorem is referenced by:  xp11m  5026
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