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Theorem ssxpb 5568
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 5553 . . . . . . . 8 |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )
2 dmxp 5344 . . . . . . . . 9 |- ( B =/= (/) -> dom ( A X. B ) = A )
32adantl 482 . . . . . . . 8 |- ( ( A =/= (/) /\ B =/= (/) ) -> dom ( A X. B ) = A )
41, 3sylbir 225 . . . . . . 7 |- ( ( A X. B ) =/= (/) -> dom ( A X. B ) = A )
54adantr 481 . . . . . 6 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> dom ( A X. B ) = A )
6 dmss 5323 . . . . . . 7 |- ( ( A X. B ) C_ ( C X. D ) -> dom ( A X. B ) C_ dom ( C X. D ) )
76adantl 482 . . . . . 6 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> dom ( A X. B ) C_ dom ( C X. D ) )
85, 7eqsstr3d 3640 . . . . 5 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ dom ( C X. D ) )
9 dmxpss 5565 . . . . 5 |- dom ( C X. D ) C_ C
108, 9syl6ss 3615 . . . 4 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> A C_ C )
11 rnxp 5564 . . . . . . . . 9 |- ( A =/= (/) -> ran ( A X. B ) = B )
1211adantr 481 . . . . . . . 8 |- ( ( A =/= (/) /\ B =/= (/) ) -> ran ( A X. B ) = B )
131, 12sylbir 225 . . . . . . 7 |- ( ( A X. B ) =/= (/) -> ran ( A X. B ) = B )
1413adantr 481 . . . . . 6 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> ran ( A X. B ) = B )
15 rnss 5354 . . . . . . 7 |- ( ( A X. B ) C_ ( C X. D ) -> ran ( A X. B ) C_ ran ( C X. D ) )
1615adantl 482 . . . . . 6 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> ran ( A X. B ) C_ ran ( C X. D ) )
1714, 16eqsstr3d 3640 . . . . 5 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ ran ( C X. D ) )
18 rnxpss 5566 . . . . 5 |- ran ( C X. D ) C_ D
1917, 18syl6ss 3615 . . . 4 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> B C_ D )
2010, 19jca 554 . . 3 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) C_ ( C X. D ) ) -> ( A C_ C /\ B C_ D ) )
2120ex 450 . 2 |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) C_ ( C X. D ) -> ( A C_ C /\ B C_ D ) ) )
22 xpss12 5225 . 2 |- ( ( A C_ C /\ B C_ D ) -> ( A X. B ) C_ ( C X. D ) )
2321, 22impbid1 215 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 196   /\ wa 384   = wceq 1483   =/= wne 2794   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114   ran crn 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125
This theorem is referenced by:  xp11  5569  dibord  36448
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