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Theorem fvixp 6821
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 6820 . . 3  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
21simp3bi 972 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
3 fveq2 5525 . . . 4  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
4 fvixp.1 . . . 4  |-  ( x  =  C  ->  B  =  D )
53, 4eleq12d 2351 . . 3  |-  ( x  =  C  ->  (
( F `  x
)  e.  B  <->  ( F `  C )  e.  D
) )
65rspccva 2883 . 2  |-  ( ( A. x  e.  A  ( F `  x )  e.  B  /\  C  e.  A )  ->  ( F `  C )  e.  D )
72, 6sylan 457 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  funcf2  13742  funcpropd  13774  natcl  13827  natpropd  13850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ixp 6818
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