Theorem nffv 5328

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 5036	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2229	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 3895	. . 3 |- F/ x A F y
6	5	nfiotaxy 4997	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2226	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2216   class class class wbr 3851   iotacio 4991   ` cfv 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036

This theorem is referenced by:  nffvmpt1  5329  nffvd  5330  dffn5imf  5372  fvmptssdm  5400  fvmptf  5408  eqfnfv2f  5415  ralrnmpt  5455  rexrnmpt  5456  ffnfvf  5471  funiunfvdmf  5557  dff13f  5563  nfiso  5599  nfrecs  6086  nffrec  6175  nfiseq  9929  seq3f1olemstep  9991  seq3f1olemp  9992  nfsum1  10806  nfsum  10807  fsumrelem  10926