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Theorem xpexr2 6717
Description: If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
xpexr2  |-  ( ( ( A  X.  B
)  e.  C  /\  ( A  X.  B
)  =/=  (/) )  -> 
( A  e.  _V  /\  B  e.  _V )
)

Proof of Theorem xpexr2
StepHypRef Expression
1 xpnz 5419 . 2  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 dmxp 5214 . . . . . 6  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32adantl 466 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  =  A )
4 dmexg 6707 . . . . . 6  |-  ( ( A  X.  B )  e.  C  ->  dom  ( A  X.  B
)  e.  _V )
54adantr 465 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  e.  _V )
63, 5eqeltrrd 2551 . . . 4  |-  ( ( ( A  X.  B
)  e.  C  /\  B  =/=  (/) )  ->  A  e.  _V )
7 rnxp 5430 . . . . . 6  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
87adantl 466 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  A  =/=  (/) )  ->  ran  ( A  X.  B
)  =  B )
9 rnexg 6708 . . . . . 6  |-  ( ( A  X.  B )  e.  C  ->  ran  ( A  X.  B
)  e.  _V )
109adantr 465 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  A  =/=  (/) )  ->  ran  ( A  X.  B
)  e.  _V )
118, 10eqeltrrd 2551 . . . 4  |-  ( ( ( A  X.  B
)  e.  C  /\  A  =/=  (/) )  ->  B  e.  _V )
126, 11anim12dan 834 . . 3  |-  ( ( ( A  X.  B
)  e.  C  /\  ( B  =/=  (/)  /\  A  =/=  (/) ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
1312ancom2s 800 . 2  |-  ( ( ( A  X.  B
)  e.  C  /\  ( A  =/=  (/)  /\  B  =/=  (/) ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
141, 13sylan2br 476 1  |-  ( ( ( A  X.  B
)  e.  C  /\  ( A  X.  B
)  =/=  (/) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   _Vcvv 3108   (/)c0 3780    X. cxp 4992   dom cdm 4994   ran crn 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rn 5005
This theorem is referenced by:  xpfir  7734  bj-xpnzex  33474
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