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Theorem unxpwdom 8048
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 7559 . . . . 5  |-  Rel  ~<_
21brrelex2i 4864 . . . 4  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( B  u.  C )  e.  _V )
3 domeng 7567 . . . 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 241 . 2  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  E. x
( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C ) ) )
6 simprl 756 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  X.  A )  ~~  x
)
7 indi 3695 . . . . . 6  |-  ( x  i^i  ( B  u.  C ) )  =  ( ( x  i^i 
B )  u.  (
x  i^i  C )
)
8 simprr 758 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  x  C_  ( B  u.  C )
)
9 df-ss 3427 . . . . . . 7  |-  ( x 
C_  ( B  u.  C )  <->  ( x  i^i  ( B  u.  C
) )  =  x )
108, 9sylib 196 . . . . . 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 2457 . . . . 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 4420 . . . 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 8047 . . . 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 3605 . . . . . . . 8  |-  B  C_  ( B  u.  C
)
162adantr 463 . . . . . . . 8  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( B  u.  C )  e.  _V )
17 ssexg 4539 . . . . . . . 8  |-  ( ( B  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  B  e.  _V )
1815, 16, 17sylancr 661 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  B  e.  _V )
19 inss2 3659 . . . . . . 7  |-  ( x  i^i  B )  C_  B
20 ssdomg 7598 . . . . . . 7  |-  ( B  e.  _V  ->  (
( x  i^i  B
)  C_  B  ->  ( x  i^i  B )  ~<_  B ) )
2118, 19, 20mpisyl 19 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_  B )
22 domwdom 8033 . . . . . 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 8034 . . . . . 6  |-  ( ( A  ~<_*  ( x  i^i  B
)  /\  ( x  i^i  B )  ~<_*  B )  ->  A  ~<_*  B )
2524expcom 433 . . . . 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 3606 . . . . . . 7  |-  C  C_  ( B  u.  C
)
28 ssexg 4539 . . . . . . 7  |-  ( ( C  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  C  e.  _V )
2927, 16, 28sylancr 661 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  C  e.  _V )
30 inss2 3659 . . . . . 6  |-  ( x  i^i  C )  C_  C
31 ssdomg 7598 . . . . . 6  |-  ( C  e.  _V  ->  (
( x  i^i  C
)  C_  C  ->  ( x  i^i  C )  ~<_  C ) )
3229, 30, 31mpisyl 19 . . . . 5  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
C )  ~<_  C )
33 domtr 7605 . . . . . 6  |-  ( ( A  ~<_  ( x  i^i 
C )  /\  (
x  i^i  C )  ~<_  C )  ->  A  ~<_  C )
3433expcom 433 . . . . 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 839 . . 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 1747 1  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    u. cun 3411    i^i cin 3412    C_ wss 3413   class class class wbr 4394    X. cxp 4820    ~~ cen 7550    ~<_ cdom 7551    ~<_* cwdom 8016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-1st 6783  df-2nd 6784  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-wdom 8018
This theorem is referenced by:  pwcdadom  8627
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