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Theorem ssxpbm 5065
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 5051 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. x  x  e.  ( A  X.  B
) )
2 dmxpm 4848 . . . . . . . . 9  |-  ( E. b  b  e.  B  ->  dom  ( A  X.  B )  =  A )
32adantl 277 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  ->  dom  ( A  X.  B )  =  A )
41, 3sylbir 135 . . . . . . 7  |-  ( E. x  x  e.  ( A  X.  B )  ->  dom  ( A  X.  B )  =  A )
54adantr 276 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  dom  ( A  X.  B )  =  A )
6 dmss 4827 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 277 . . . . . 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 3193 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  dom  ( C  X.  D
) )
9 dmxpss 5060 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9sstrdi 3168 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  C
)
11 rnxpm 5059 . . . . . . . . 9  |-  ( E. a  a  e.  A  ->  ran  ( A  X.  B )  =  B )
1211adantr 276 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  ->  ran  ( A  X.  B )  =  B )
131, 12sylbir 135 . . . . . . 7  |-  ( E. x  x  e.  ( A  X.  B )  ->  ran  ( A  X.  B )  =  B )
1413adantr 276 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ran  ( A  X.  B )  =  B )
15 rnss 4858 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 277 . . . . . 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 3193 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  ran  ( C  X.  D
) )
18 rnxpss 5061 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18sstrdi 3168 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  D
)
2010, 19jca 306 . . 3  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ( A  C_  C  /\  B  C_  D
) )
2120ex 115 . 2  |-  ( E. x  x  e.  ( A  X.  B )  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
22 xpss12 4734 . 2  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( A  X.  B
)  C_  ( C  X.  D ) )
2321, 22impbid1 142 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 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148    C_ wss 3130    X. cxp 4625   dom cdm 4627   ran crn 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-dm 4637  df-rn 4638
This theorem is referenced by:  xp11m  5068
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