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Theorem wdom2d2 35596
Description: Deduction for weak dominance by a Cartesian 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 6607 . . 3  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  X.  C
)  e.  _V )
52, 3, 4syl2anc 665 . 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 3417 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
87nfeq2 2608 . . . 4  |-  F/ y  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
9 nfcv 2591 . . . . . 6  |-  F/_ z
( 1st `  w
)
10 nfcsb1v 3417 . . . . . 6  |-  F/_ z [_ ( 2nd `  w
)  /  z ]_ X
119, 10nfcsb 3419 . . . . 5  |-  F/_ z [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
1211nfeq2 2608 . . . 4  |-  F/ z  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
13 nfv 1754 . . . 4  |-  F/ w  x  =  X
14 csbopeq1a 6860 . . . . 5  |-  ( w  =  <. y ,  z
>.  ->  [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X  =  X )
1514eqeq2d 2443 . . . 4  |-  ( w  =  <. y ,  z
>.  ->  ( x  = 
[_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  x  =  X ) )
168, 12, 13, 15rexxpf 5002 . . 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 215 . 2  |-  ( (
ph  /\  x  e.  A )  ->  E. w  e.  ( B  X.  C
) x  =  [_ ( 1st `  w )  /  y ]_ [_ ( 2nd `  w )  / 
z ]_ X )
181, 5, 17wdom2d 8095 1  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   _Vcvv 3087   [_csb 3401   <.cop 4008   class class class wbr 4426    X. cxp 4852   ` cfv 5601   1stc1st 6805   2ndc2nd 6806    ~<_* cwdom 8072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-wdom 8074
This theorem is referenced by: (None)
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