Theorem nffv 5685

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 5365	. 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 4161	. . 3 |- F/ x A F y
6	5	nfiotaw 5321	. 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 4114   iotacio 5315   ` cfv 5357

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 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365

This theorem is referenced by:  nffvmpt1  5686  nffvd  5687  dffn5imf  5737  fvmptssdm  5767  fvmptf  5775  eqfnfv2f  5784  ralrnmpt  5824  rexrnmpt  5825  ffnfvf  5841  dfimafnf  5928  funiunfvdmf  5943  dff13f  5949  nfiso  5985  nfrecs  6551  nffrec  6640  cc2  7597  nfseq  10843  seq3f1olemstep  10900  seq3f1olemp  10901  nfsum1  12066  nfsum  12067  fsumrelem  12182  nfcprod1  12265  nfcprod  12266  ctiunctlemfo  13274  ctiunct  13275  prdsbas3  14129  cnmpt11  15274  cnmpt21  15282  lgseisenlem2  16070