Theorem csbfvg 5555

Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.)

Assertion

Ref	Expression
csbfvg	|- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )

Distinct variable group:   x, F

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem csbfvg

Step	Hyp	Ref	Expression
1		csbfv2g 5554	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` [_ A / x ]_ x ) )
2		csbvarg 3087	. . 3 |- ( A e. C -> [_ A / x ]_ x = A )
3	2	fveq2d 5521	. 2 |- ( A e. C -> ( F ` [_ A / x ]_ x ) = ( F ` A ) )
4	1, 3	eqtrd 2210	1 |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   [_csb 3059   ` cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226

This theorem is referenced by:  ixpsnval  6703