Theorem nffv 5205

Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nffv.1	|- F/_ x F
nffv.2	|- F/_ x A

Assertion

Ref	Expression
nffv	|- F/_ x ( F ` A )

Proof of Theorem nffv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 4930	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2219	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 3829	. . 3 |- F/ x A F y
6	5	nfiotaxy 4891	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2216	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2206   class class class wbr 3785   iotacio 4885   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930

This theorem is referenced by:  nffvmpt1  5206  nffvd  5207  dffn5imf  5249  fvmptssdm  5276  fvmptf  5284  eqfnfv2f  5290  ralrnmpt  5330  rexrnmpt  5331  ffnfvf  5345  funiunfvdmf  5424  dff13f  5430  nfiso  5466  nfrecs  5945  nffrec  6005  nfiseq  9438  nfsum1  10193  nfsum  10194