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Theorem unxpwdom 7271
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
StepHypRef Expression
1 reldom 6837 . . . . 5  |-  Rel  ~<_
21brrelex2i 4718 . . . 4  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( B  u.  C )  e.  _V )
3 domeng 6844 . . . 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 234 . 2  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  E. x
( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C ) ) )
6 simprl 735 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  X.  A )  ~~  x
)
7 indi 3390 . . . . . . . 8  |-  ( x  i^i  ( B  u.  C ) )  =  ( ( x  i^i 
B )  u.  (
x  i^i  C )
)
8 simprr 736 . . . . . . . . 9  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  x  C_  ( B  u.  C )
)
9 df-ss 3141 . . . . . . . . 9  |-  ( x 
C_  ( B  u.  C )  <->  ( x  i^i  ( B  u.  C
) )  =  x )
108, 9sylib 190 . . . . . . . 8  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i  ( B  u.  C
) )  =  x )
117, 10syl5eqr 2304 . . . . . . 7  |-  ( ( ( 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 4023 . . . . . 6  |-  ( ( ( 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 7270 . . . . . 6  |-  ( ( 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 . . . . 5  |-  ( ( ( 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 3313 . . . . . . . . . 10  |-  B  C_  ( B  u.  C
)
162adantr 453 . . . . . . . . . 10  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( B  u.  C )  e.  _V )
17 ssexg 4134 . . . . . . . . . 10  |-  ( ( B  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  B  e.  _V )
1815, 16, 17sylancr 647 . . . . . . . . 9  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  B  e.  _V )
19 inss2 3365 . . . . . . . . 9  |-  ( x  i^i  B )  C_  B
20 ssdomg 6875 . . . . . . . . 9  |-  ( B  e.  _V  ->  (
( x  i^i  B
)  C_  B  ->  ( x  i^i  B )  ~<_  B ) )
2118, 19, 20ee10 1372 . . . . . . . 8  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_  B )
22 domwdom 7256 . . . . . . . 8  |-  ( ( x  i^i  B )  ~<_  B  ->  ( x  i^i  B )  ~<_*  B )
2321, 22syl 17 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
B )  ~<_*  B )
24 wdomtr 7257 . . . . . . . 8  |-  ( ( A  ~<_*  ( x  i^i  B
)  /\  ( x  i^i  B )  ~<_*  B )  ->  A  ~<_*  B )
2524expcom 426 . . . . . . 7  |-  ( ( x  i^i  B )  ~<_*  B  ->  ( A  ~<_*  (
x  i^i  B )  ->  A  ~<_*  B ) )
2623, 25syl 17 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_*  (
x  i^i  B )  ->  A  ~<_*  B ) )
27 ssun2 3314 . . . . . . . . 9  |-  C  C_  ( B  u.  C
)
28 ssexg 4134 . . . . . . . . 9  |-  ( ( C  C_  ( B  u.  C )  /\  ( B  u.  C )  e.  _V )  ->  C  e.  _V )
2927, 16, 28sylancr 647 . . . . . . . 8  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  C  e.  _V )
30 inss2 3365 . . . . . . . 8  |-  ( x  i^i  C )  C_  C
31 ssdomg 6875 . . . . . . . 8  |-  ( C  e.  _V  ->  (
( x  i^i  C
)  C_  C  ->  ( x  i^i  C )  ~<_  C ) )
3229, 30, 31ee10 1372 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( x  i^i 
C )  ~<_  C )
33 domtr 6882 . . . . . . . 8  |-  ( ( A  ~<_  ( x  i^i 
C )  /\  (
x  i^i  C )  ~<_  C )  ->  A  ~<_  C )
3433expcom 426 . . . . . . 7  |-  ( ( x  i^i  C )  ~<_  C  ->  ( A  ~<_  ( x  i^i  C )  ->  A  ~<_  C ) )
3532, 34syl 17 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_  ( x  i^i  C )  ->  A  ~<_  C ) )
3626, 35orim12d 814 . . . . 5  |-  ( ( ( 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 16 . . . 4  |-  ( ( ( A  X.  A
)  ~<_  ( B  u.  C )  /\  (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) ) )  ->  ( A  ~<_*  B  \/  A  ~<_  C )
)
3837ex 425 . . 3  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( (
( A  X.  A
)  ~~  x  /\  x  C_  ( B  u.  C ) )  -> 
( A  ~<_*  B  \/  A  ~<_  C ) ) )
3938exlimdv 1933 . 2  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( E. x ( ( A  X.  A )  ~~  x  /\  x  C_  ( B  u.  C )
)  ->  ( A  ~<_*  B  \/  A  ~<_  C ) ) )
405, 39mpd 16 1  |-  ( ( A  X.  A )  ~<_  ( B  u.  C
)  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2763    u. cun 3125    i^i cin 3126    C_ wss 3127   class class class wbr 3997    X. cxp 4659    ~~ cen 6828    ~<_ cdom 6829    ~<_* cwdom 7239
This theorem is referenced by:  pwcdadom  7810
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1st 6056  df-2nd 6057  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-wdom 7241
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