Theorem nffv 5637

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 5326	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2372	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4130	. . 3 |- F/ x A F y
6	5	nfiotaw 5282	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2369	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2359   class class class wbr 4083   iotacio 5276   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326

This theorem is referenced by:  nffvmpt1  5638  nffvd  5639  dffn5imf  5689  fvmptssdm  5719  fvmptf  5727  eqfnfv2f  5736  ralrnmpt  5777  rexrnmpt  5778  ffnfvf  5794  funiunfvdmf  5888  dff13f  5894  nfiso  5930  nfrecs  6453  nffrec  6542  cc2  7453  nfseq  10679  seq3f1olemstep  10736  seq3f1olemp  10737  nfsum1  11867  nfsum  11868  fsumrelem  11982  nfcprod1  12065  nfcprod  12066  ctiunctlemfo  13010  ctiunct  13011  prdsbas3  13320  cnmpt11  14957  cnmpt21  14965  lgseisenlem2  15750