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Theorem ssxpbm 4806
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 4795 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. x  x  e.  ( A  X.  B
) )
2 dmxpm 4603 . . . . . . . . 9  |-  ( E. b  b  e.  B  ->  dom  ( A  X.  B )  =  A )
32adantl 271 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  ->  dom  ( A  X.  B )  =  A )
41, 3sylbir 133 . . . . . . 7  |-  ( E. x  x  e.  ( A  X.  B )  ->  dom  ( A  X.  B )  =  A )
54adantr 270 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  dom  ( A  X.  B )  =  A )
6 dmss 4582 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 271 . . . . . 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, 7eqsstr3d 3043 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  dom  ( C  X.  D
) )
9 dmxpss 4803 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9syl6ss 3020 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  C
)
11 rnxpm 4802 . . . . . . . . 9  |-  ( E. a  a  e.  A  ->  ran  ( A  X.  B )  =  B )
1211adantr 270 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  ->  ran  ( A  X.  B )  =  B )
131, 12sylbir 133 . . . . . . 7  |-  ( E. x  x  e.  ( A  X.  B )  ->  ran  ( A  X.  B )  =  B )
1413adantr 270 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ran  ( A  X.  B )  =  B )
15 rnss 4612 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 271 . . . . . 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, 16eqsstr3d 3043 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  ran  ( C  X.  D
) )
18 rnxpss 4804 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18syl6ss 3020 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  D
)
2010, 19jca 300 . . 3  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ( A  C_  C  /\  B  C_  D
) )
2120ex 113 . 2  |-  ( E. x  x  e.  ( A  X.  B )  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
22 xpss12 4493 . 2  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( A  X.  B
)  C_  ( C  X.  D ) )
2321, 22impbid1 140 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 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2982    X. cxp 4389   dom cdm 4391   ran crn 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  xp11m  4809
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