Theorem nffv 5705

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 5385	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2392	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4177	. . 3 |- F/ x A F y
6	5	nfiotaw 5341	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2389	1 |- F/_ x ( F ` A )

This proof depends on syntax axioms:   F/_wnfc 2379   class class class wbr 4130   iotacio 5335   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385

This theorem is used by:  nffvmpt1  5706  nffvd  5707  dffn5imf  5758  fvmptssdm  5790  fvmptf  5798  eqfnfv2f  5810  ralrnmpt  5850  rexrnmpt  5851  ffnfvf  5867  dfimafnf  5955  funiunfvdmf  5970  dff13f  5976  nfiso  6012  nfrecs  6578  nffrec  6667  cc2  7633  nfseq  10894  seq3f1olemstep  10951  seq3f1olemp  10952  nfsum1  12122  nfsum  12123  fsumrelem  12238  nfcprod1  12321  nfcprod  12322  ctiunctlemfo  13330  ctiunct  13331  prdsbas3  14187  cnmpt11  15384  cnmpt21  15392  lgseisenlem2  16190