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Theorem wdom2d2 30808
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 6587 . . 3  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  X.  C
)  e.  _V )
52, 3, 4syl2anc 661 . 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 3451 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
87nfeq2 2646 . . . 4  |-  F/ y  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
9 nfcv 2629 . . . . . 6  |-  F/_ z
( 1st `  w
)
10 nfcsb1v 3451 . . . . . 6  |-  F/_ z [_ ( 2nd `  w
)  /  z ]_ X
119, 10nfcsb 3453 . . . . 5  |-  F/_ z [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
1211nfeq2 2646 . . . 4  |-  F/ z  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
13 nfv 1683 . . . 4  |-  F/ w  x  =  X
14 csbopeq1a 6838 . . . . 5  |-  ( w  =  <. y ,  z
>.  ->  [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X  =  X )
1514eqeq2d 2481 . . . 4  |-  ( w  =  <. y ,  z
>.  ->  ( x  = 
[_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  x  =  X ) )
168, 12, 13, 15rexxpf 5150 . . 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 212 . 2  |-  ( (
ph  /\  x  e.  A )  ->  E. w  e.  ( B  X.  C
) x  =  [_ ( 1st `  w )  /  y ]_ [_ ( 2nd `  w )  / 
z ]_ X )
181, 5, 17wdom2d 8007 1  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   [_csb 3435   <.cop 4033   class class class wbr 4447    X. cxp 4997   ` cfv 5588   1stc1st 6783   2ndc2nd 6784    ~<_* cwdom 7984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1st 6785  df-2nd 6786  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-wdom 7986
This theorem is referenced by: (None)
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