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Theorem fvixp 7030
Description: Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
fvixp.1  |-  ( x  =  C  ->  B  =  D )
Assertion
Ref Expression
fvixp  |-  ( ( F  e.  X_ x  e.  A  B  /\  C  e.  A )  ->  ( F `  C
)  e.  D )
Distinct variable groups:    x, A    x, C    x, D    x, F
Allowed substitution hint:    B( x)

Proof of Theorem fvixp
StepHypRef Expression
1 elixp2 7029 . . 3  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
21simp3bi 974 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
3 fveq2 5691 . . . 4  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
4 fvixp.1 . . . 4  |-  ( x  =  C  ->  B  =  D )
53, 4eleq12d 2476 . . 3  |-  ( x  =  C  ->  (
( F `  x
)  e.  B  <->  ( F `  C )  e.  D
) )
65rspccva 3015 . 2  |-  ( ( A. x  e.  A  ( F `  x )  e.  B  /\  C  e.  A )  ->  ( F `  C )  e.  D )
72, 6sylan 458 1  |-  ( ( F  e.  X_ x  e.  A  B  /\  C  e.  A )  ->  ( F `  C
)  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920    Fn wfn 5412   ` cfv 5417   X_cixp 7026
This theorem is referenced by:  funcf2  14024  funcpropd  14056  natcl  14109  natpropd  14132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fn 5420  df-fv 5425  df-ixp 7027
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