Theorem nffv 5571

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 5267	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2339	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4080	. . 3 |- F/ x A F y
6	5	nfiotaw 5224	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2336	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2326   class class class wbr 4034   iotacio 5218   ` cfv 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267

This theorem is referenced by:  nffvmpt1  5572  nffvd  5573  dffn5imf  5619  fvmptssdm  5649  fvmptf  5657  eqfnfv2f  5666  ralrnmpt  5707  rexrnmpt  5708  ffnfvf  5724  funiunfvdmf  5814  dff13f  5820  nfiso  5856  nfrecs  6374  nffrec  6463  cc2  7350  nfseq  10566  seq3f1olemstep  10623  seq3f1olemp  10624  nfsum1  11538  nfsum  11539  fsumrelem  11653  nfcprod1  11736  nfcprod  11737  ctiunctlemfo  12681  ctiunct  12682  prdsbas3  12989  cnmpt11  14603  cnmpt21  14611  lgseisenlem2  15396