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Theorem fvixp 7467
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 7466 . . 3  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
21simp3bi 1011 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
3 fveq2 5848 . . . 4  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
4 fvixp.1 . . . 4  |-  ( x  =  C  ->  B  =  D )
53, 4eleq12d 2536 . . 3  |-  ( x  =  C  ->  (
( F `  x
)  e.  B  <->  ( F `  C )  e.  D
) )
65rspccva 3206 . 2  |-  ( ( A. x  e.  A  ( F `  x )  e.  B  /\  C  e.  A )  ->  ( F `  C )  e.  D )
72, 6sylan 469 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    Fn wfn 5565   ` cfv 5570   X_cixp 7462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-ixp 7463
This theorem is referenced by:  funcf2  15359  funcpropd  15391  natcl  15444  natpropd  15467  finixpnum  30281
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