MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvixp Unicode version

Theorem fvixp 6909
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 6908 . . 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 5608 . . . 4  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
4 fvixp.1 . . . 4  |-  ( x  =  C  ->  B  =  D )
53, 4eleq12d 2426 . . 3  |-  ( x  =  C  ->  (
( F `  x
)  e.  B  <->  ( F `  C )  e.  D
) )
65rspccva 2959 . 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 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    Fn wfn 5332   ` cfv 5337   X_cixp 6905
This theorem is referenced by:  funcf2  13841  funcpropd  13873  natcl  13926  natpropd  13949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345  df-ixp 6906
  Copyright terms: Public domain W3C validator