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Theorem wdom2d2 27139
Description: Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
Hypotheses
Ref Expression
wdom2d2.a  |-  ( ph  ->  A  e.  V )
wdom2d2.b  |-  ( ph  ->  B  e.  W )
wdom2d2.c  |-  ( ph  ->  C  e.  X )
wdom2d2.o  |-  ( (
ph  /\  x  e.  A )  ->  E. y  e.  B  E. z  e.  C  x  =  X )
Assertion
Ref Expression
wdom2d2  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Distinct variable groups:    x, y,
z    x, X    x, A    x, B, y    x, C, y, z    ph, x
Allowed substitution hints:    ph( y, z)    A( y, z)    B( z)    V( x, y, z)    W( x, y, z)    X( y, z)

Proof of Theorem wdom2d2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 wdom2d2.a . 2  |-  ( ph  ->  A  e.  V )
2 wdom2d2.b . . 3  |-  ( ph  ->  B  e.  W )
3 wdom2d2.c . . 3  |-  ( ph  ->  C  e.  X )
4 xpexg 4802 . . 3  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  X.  C
)  e.  _V )
52, 3, 4syl2anc 642 . 2  |-  ( ph  ->  ( B  X.  C
)  e.  _V )
6 wdom2d2.o . . 3  |-  ( (
ph  /\  x  e.  A )  ->  E. y  e.  B  E. z  e.  C  x  =  X )
7 nfcsb1v 3115 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
87nfeq2 2432 . . . 4  |-  F/ y  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
9 nfcv 2421 . . . . . 6  |-  F/_ z
( 1st `  w
)
10 nfcsb1v 3115 . . . . . 6  |-  F/_ z [_ ( 2nd `  w
)  /  z ]_ X
119, 10nfcsb 3117 . . . . 5  |-  F/_ z [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
1211nfeq2 2432 . . . 4  |-  F/ z  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
13 nfv 1607 . . . 4  |-  F/ w  x  =  X
14 csbopeq1a 6175 . . . . 5  |-  ( w  =  <. y ,  z
>.  ->  [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X  =  X )
1514eqeq2d 2296 . . . 4  |-  ( w  =  <. y ,  z
>.  ->  ( x  = 
[_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  x  =  X ) )
168, 12, 13, 15rexxpf 4833 . . 3  |-  ( E. w  e.  ( B  X.  C ) x  =  [_ ( 1st `  w )  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  E. y  e.  B  E. z  e.  C  x  =  X )
176, 16sylibr 203 . 2  |-  ( (
ph  /\  x  e.  A )  ->  E. w  e.  ( B  X.  C
) x  =  [_ ( 1st `  w )  /  y ]_ [_ ( 2nd `  w )  / 
z ]_ X )
181, 5, 17wdom2d 7296 1  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   E.wrex 2546   _Vcvv 2790   [_csb 3083   <.cop 3645   class class class wbr 4025    X. cxp 4689   ` cfv 5257   1stc1st 6122   2ndc2nd 6123    ~<_* cwdom 7273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-1st 6124  df-2nd 6125  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-wdom 7275
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