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Theorem wdom2d2 26276
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 4837 . . 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 3147 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
87nfeq2 2463 . . . 4  |-  F/ y  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
9 nfcv 2452 . . . . . 6  |-  F/_ z
( 1st `  w
)
10 nfcsb1v 3147 . . . . . 6  |-  F/_ z [_ ( 2nd `  w
)  /  z ]_ X
119, 10nfcsb 3149 . . . . 5  |-  F/_ z [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
1211nfeq2 2463 . . . 4  |-  F/ z  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
13 nfv 1610 . . . 4  |-  F/ w  x  =  X
14 csbopeq1a 6215 . . . . 5  |-  ( w  =  <. y ,  z
>.  ->  [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X  =  X )
1514eqeq2d 2327 . . . 4  |-  ( w  =  <. y ,  z
>.  ->  ( x  = 
[_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  x  =  X ) )
168, 12, 13, 15rexxpf 4868 . . 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 7339 1  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578   _Vcvv 2822   [_csb 3115   <.cop 3677   class class class wbr 4060    X. cxp 4724   ` cfv 5292   1stc1st 6162   2ndc2nd 6163    ~<_* cwdom 7316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-1st 6164  df-2nd 6165  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-wdom 7318
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