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Theorem ssxpbm 4942
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 4928 . . . . . . . 8  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. x  x  e.  ( A  X.  B
) )
2 dmxpm 4727 . . . . . . . . 9  |-  ( E. b  b  e.  B  ->  dom  ( A  X.  B )  =  A )
32adantl 273 . . . . . . . 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 272 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  dom  ( A  X.  B )  =  A )
6 dmss 4706 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  dom  ( A  X.  B
)  C_  dom  ( C  X.  D ) )
76adantl 273 . . . . . 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 3102 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  dom  ( C  X.  D
) )
9 dmxpss 4937 . . . . 5  |-  dom  ( C  X.  D )  C_  C
108, 9syl6ss 3077 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  A  C_  C
)
11 rnxpm 4936 . . . . . . . . 9  |-  ( E. a  a  e.  A  ->  ran  ( A  X.  B )  =  B )
1211adantr 272 . . . . . . . 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 272 . . . . . 6  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  ran  ( A  X.  B )  =  B )
15 rnss 4737 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( C  X.  D )  ->  ran  ( A  X.  B
)  C_  ran  ( C  X.  D ) )
1615adantl 273 . . . . . 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 3102 . . . . 5  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  ran  ( C  X.  D
) )
18 rnxpss 4938 . . . . 5  |-  ran  ( C  X.  D )  C_  D
1917, 18syl6ss 3077 . . . 4  |-  ( ( E. x  x  e.  ( A  X.  B
)  /\  ( A  X.  B )  C_  ( C  X.  D ) )  ->  B  C_  D
)
2010, 19jca 302 . . 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 4614 . 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 1314   E.wex 1451    e. wcel 1463    C_ wss 3039    X. cxp 4505   dom cdm 4507   ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  xp11m  4945
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