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Theorem unxpwdom 8108
Description: If a Cartesian 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 7581 . . . . 5  |-  Rel  ~<_
21brrelex2i 4893 . . . 4  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( B  u.  C )  e.  _V )
3 domeng 7589 . . . 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 17 . . 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 245 . 2  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  E. x
( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C ) ) )
6 simprl 763 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  X.  A )  ~~  x
)
7 indi 3720 . . . . . 6  |-  ( x  i^i  ( B  u.  C ) )  =  ( ( x  i^i 
B )  u.  (
x  i^i  C )
)
8 simprr 765 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  x  C_  ( B  u.  C )
)
9 df-ss 3451 . . . . . . 7  |-  ( x 
C_  ( B  u.  C )  <->  ( x  i^i  ( B  u.  C
) )  =  x )
108, 9sylib 200 . . . . . 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 2478 . . . . 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 4448 . . . 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 8107 . . . 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 17 . . 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 3630 . . . . . . . 8  |-  B  C_  ( B  u.  C
)
162adantr 467 . . . . . . . 8  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( B  u.  C )  e.  _V )
17 ssexg 4568 . . . . . . . 8  |-  ( ( B  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  B  e.  _V )
1815, 16, 17sylancr 668 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  B  e.  _V )
19 inss2 3684 . . . . . . 7  |-  ( x  i^i  B )  C_  B
20 ssdomg 7620 . . . . . . 7  |-  ( B  e.  _V  ->  (
( x  i^i  B
)  C_  B  ->  ( x  i^i  B )  ~<_  B ) )
2118, 19, 20mpisyl 22 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_  B )
22 domwdom 8093 . . . . . 6  |-  ( ( x  i^i  B )  ~<_  B  ->  ( x  i^i  B )  ~<_*  B )
2321, 22syl 17 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_*  B )
24 wdomtr 8094 . . . . . 6  |-  ( ( A  ~<_*  ( x  i^i  B
)  /\  ( x  i^i  B )  ~<_*  B )  ->  A  ~<_*  B )
2524expcom 437 . . . . 5  |-  ( ( x  i^i  B )  ~<_*  B  ->  ( A  ~<_*  (
x  i^i  B )  ->  A  ~<_*  B ) )
2623, 25syl 17 . . . 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 3631 . . . . . . 7  |-  C  C_  ( B  u.  C
)
28 ssexg 4568 . . . . . . 7  |-  ( ( C  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  C  e.  _V )
2927, 16, 28sylancr 668 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  C  e.  _V )
30 inss2 3684 . . . . . 6  |-  ( x  i^i  C )  C_  C
31 ssdomg 7620 . . . . . 6  |-  ( C  e.  _V  ->  (
( x  i^i  C
)  C_  C  ->  ( x  i^i  C )  ~<_  C ) )
3229, 30, 31mpisyl 22 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
C )  ~<_  C )
33 domtr 7627 . . . . . 6  |-  ( ( A  ~<_  ( x  i^i 
C )  /\  (
x  i^i  C )  ~<_  C )  ->  A  ~<_  C )
3433expcom 437 . . . . 5  |-  ( ( x  i^i  C )  ~<_  C  ->  ( A  ~<_  ( x  i^i  C )  ->  A  ~<_  C ) )
3532, 34syl 17 . . . 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 847 . . 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 1771 1  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1438   E.wex 1660    e. wcel 1869   _Vcvv 3082    u. cun 3435    i^i cin 3436    C_ wss 3437   class class class wbr 4421    X. cxp 4849    ~~ cen 7572    ~<_ cdom 7573    ~<_* cwdom 8076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-1st 6805  df-2nd 6806  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-wdom 8078
This theorem is referenced by:  pwcdadom  8648
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