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Theorem unxpwdom 7490
Description: If a cross product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
unxpwdom  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )

Proof of Theorem unxpwdom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reldom 7051 . . . . 5  |-  Rel  ~<_
21brrelex2i 4859 . . . 4  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( B  u.  C )  e.  _V )
3 domeng 7058 . . . 4  |-  ( ( B  u.  C )  e.  _V  ->  (
( A  X.  A
)  ~<_  ( B  u.  C )  <->  E. x
( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C ) ) ) )
42, 3syl 16 . . 3  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( ( A  X.  A )  ~<_  ( B  u.  C )  <->  E. x ( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C )
) ) )
54ibi 233 . 2  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  E. x
( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C ) ) )
6 simprl 733 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  X.  A )  ~~  x
)
7 indi 3530 . . . . . 6  |-  ( x  i^i  ( B  u.  C ) )  =  ( ( x  i^i 
B )  u.  (
x  i^i  C )
)
8 simprr 734 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  x  C_  ( B  u.  C )
)
9 df-ss 3277 . . . . . . 7  |-  ( x 
C_  ( B  u.  C )  <->  ( x  i^i  ( B  u.  C
) )  =  x )
108, 9sylib 189 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i  ( B  u.  C
) )  =  x )
117, 10syl5eqr 2433 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( ( x  i^i  B )  u.  ( x  i^i  C
) )  =  x )
126, 11breqtrrd 4179 . . . 4  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  X.  A )  ~~  (
( x  i^i  B
)  u.  ( x  i^i  C ) ) )
13 unxpwdom2 7489 . . . 4  |-  ( ( A  X.  A ) 
~~  ( ( x  i^i  B )  u.  ( x  i^i  C
) )  ->  ( A  ~<_*  ( x  i^i  B
)  \/  A  ~<_  ( x  i^i  C ) ) )
1412, 13syl 16 . . 3  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_*  (
x  i^i  B )  \/  A  ~<_  ( x  i^i  C ) ) )
15 ssun1 3453 . . . . . . . 8  |-  B  C_  ( B  u.  C
)
162adantr 452 . . . . . . . 8  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( B  u.  C )  e.  _V )
17 ssexg 4290 . . . . . . . 8  |-  ( ( B  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  B  e.  _V )
1815, 16, 17sylancr 645 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  B  e.  _V )
19 inss2 3505 . . . . . . 7  |-  ( x  i^i  B )  C_  B
20 ssdomg 7089 . . . . . . 7  |-  ( B  e.  _V  ->  (
( x  i^i  B
)  C_  B  ->  ( x  i^i  B )  ~<_  B ) )
2118, 19, 20ee10 1382 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_  B )
22 domwdom 7475 . . . . . 6  |-  ( ( x  i^i  B )  ~<_  B  ->  ( x  i^i  B )  ~<_*  B )
2321, 22syl 16 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_*  B )
24 wdomtr 7476 . . . . . 6  |-  ( ( A  ~<_*  ( x  i^i  B
)  /\  ( x  i^i  B )  ~<_*  B )  ->  A  ~<_*  B )
2524expcom 425 . . . . 5  |-  ( ( x  i^i  B )  ~<_*  B  ->  ( A  ~<_*  (
x  i^i  B )  ->  A  ~<_*  B ) )
2623, 25syl 16 . . . 4  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_*  (
x  i^i  B )  ->  A  ~<_*  B ) )
27 ssun2 3454 . . . . . . 7  |-  C  C_  ( B  u.  C
)
28 ssexg 4290 . . . . . . 7  |-  ( ( C  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  C  e.  _V )
2927, 16, 28sylancr 645 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  C  e.  _V )
30 inss2 3505 . . . . . 6  |-  ( x  i^i  C )  C_  C
31 ssdomg 7089 . . . . . 6  |-  ( C  e.  _V  ->  (
( x  i^i  C
)  C_  C  ->  ( x  i^i  C )  ~<_  C ) )
3229, 30, 31ee10 1382 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
C )  ~<_  C )
33 domtr 7096 . . . . . 6  |-  ( ( A  ~<_  ( x  i^i 
C )  /\  (
x  i^i  C )  ~<_  C )  ->  A  ~<_  C )
3433expcom 425 . . . . 5  |-  ( ( x  i^i  C )  ~<_  C  ->  ( A  ~<_  ( x  i^i  C )  ->  A  ~<_  C ) )
3532, 34syl 16 . . . 4  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_  ( x  i^i  C )  ->  A  ~<_  C ) )
3626, 35orim12d 812 . . 3  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( ( A  ~<_*  ( x  i^i  B )  \/  A  ~<_  ( x  i^i  C ) )  ->  ( A  ~<_*  B  \/  A  ~<_  C )
) )
3714, 36mpd 15 . 2  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_*  B  \/  A  ~<_  C )
)
385, 37exlimddv 1645 1  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899    u. cun 3261    i^i cin 3262    C_ wss 3263   class class class wbr 4153    X. cxp 4816    ~~ cen 7042    ~<_ cdom 7043    ~<_* cwdom 7458
This theorem is referenced by:  pwcdadom  8029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1st 6288  df-2nd 6289  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-wdom 7460
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