Theorem nffv 5686

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 5366	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2386	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4162	. . 3 |- F/ x A F y
6	5	nfiotaw 5322	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2383	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2373   class class class wbr 4115   iotacio 5316   ` cfv 5358

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-iota 5318  df-fv 5366

This theorem is referenced by:  nffvmpt1  5687  nffvd  5688  dffn5imf  5738  fvmptssdm  5768  fvmptf  5776  eqfnfv2f  5785  ralrnmpt  5825  rexrnmpt  5826  ffnfvf  5842  dfimafnf  5929  funiunfvdmf  5944  dff13f  5950  nfiso  5986  nfrecs  6552  nffrec  6641  cc2  7598  nfseq  10847  seq3f1olemstep  10904  seq3f1olemp  10905  nfsum1  12071  nfsum  12072  fsumrelem  12187  nfcprod1  12270  nfcprod  12271  ctiunctlemfo  13279  ctiunct  13280  prdsbas3  14134  cnmpt11  15279  cnmpt21  15287  lgseisenlem2  16075