MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpexr2 Structured version   Unicode version

Theorem xpexr2 6714
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 5411 . 2  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 dmxp 5210 . . . . . 6  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32adantl 464 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  =  A )
4 dmexg 6704 . . . . . 6  |-  ( ( A  X.  B )  e.  C  ->  dom  ( A  X.  B
)  e.  _V )
54adantr 463 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  B  =/=  (/) )  ->  dom  ( A  X.  B
)  e.  _V )
63, 5eqeltrrd 2543 . . . 4  |-  ( ( ( A  X.  B
)  e.  C  /\  B  =/=  (/) )  ->  A  e.  _V )
7 rnxp 5422 . . . . . 6  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
87adantl 464 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  A  =/=  (/) )  ->  ran  ( A  X.  B
)  =  B )
9 rnexg 6705 . . . . . 6  |-  ( ( A  X.  B )  e.  C  ->  ran  ( A  X.  B
)  e.  _V )
109adantr 463 . . . . 5  |-  ( ( ( A  X.  B
)  e.  C  /\  A  =/=  (/) )  ->  ran  ( A  X.  B
)  e.  _V )
118, 10eqeltrrd 2543 . . . 4  |-  ( ( ( A  X.  B
)  e.  C  /\  A  =/=  (/) )  ->  B  e.  _V )
126, 11anim12dan 835 . . 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 474 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783    X. cxp 4986   dom cdm 4988   ran crn 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999
This theorem is referenced by:  xpfir  7735  bj-xpnzex  34936
  Copyright terms: Public domain W3C validator