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Theorem fvixp 7471
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 7470 . . 3  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
21simp3bi 1013 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
3 fveq2 5864 . . . 4  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
4 fvixp.1 . . . 4  |-  ( x  =  C  ->  B  =  D )
53, 4eleq12d 2549 . . 3  |-  ( x  =  C  ->  (
( F `  x
)  e.  B  <->  ( F `  C )  e.  D
) )
65rspccva 3213 . 2  |-  ( ( A. x  e.  A  ( F `  x )  e.  B  /\  C  e.  A )  ->  ( F `  C )  e.  D )
72, 6sylan 471 1  |-  ( ( F  e.  X_ x  e.  A  B  /\  C  e.  A )  ->  ( F `  C
)  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    Fn wfn 5581   ` cfv 5586   X_cixp 7466
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ixp 7467
This theorem is referenced by:  funcf2  15091  funcpropd  15123  natcl  15176  natpropd  15199  finixpnum  29615
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