Theorem nffv 5700

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 5380	. 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 4172	. . 3 |- F/ x A F y
6	5	nfiotaw 5336	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2389	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2379   class class class wbr 4125   iotacio 5330   ` cfv 5372

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 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 theorem 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 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  nffvmpt1  5701  nffvd  5702  dffn5imf  5752  fvmptssdm  5784  fvmptf  5792  eqfnfv2f  5801  ralrnmpt  5841  rexrnmpt  5842  ffnfvf  5858  dfimafnf  5945  funiunfvdmf  5960  dff13f  5966  nfiso  6002  nfrecs  6568  nffrec  6657  cc2  7623  nfseq  10872  seq3f1olemstep  10929  seq3f1olemp  10930  nfsum1  12100  nfsum  12101  fsumrelem  12216  nfcprod1  12299  nfcprod  12300  ctiunctlemfo  13308  ctiunct  13309  prdsbas3  14164  cnmpt11  15307  cnmpt21  15315  lgseisenlem2  16104