Theorem nffv 5586

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 5279	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2348	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4090	. . 3 |- F/ x A F y
6	5	nfiotaw 5236	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2345	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2335   class class class wbr 4044   iotacio 5230   ` cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279

This theorem is referenced by:  nffvmpt1  5587  nffvd  5588  dffn5imf  5634  fvmptssdm  5664  fvmptf  5672  eqfnfv2f  5681  ralrnmpt  5722  rexrnmpt  5723  ffnfvf  5739  funiunfvdmf  5833  dff13f  5839  nfiso  5875  nfrecs  6393  nffrec  6482  cc2  7379  nfseq  10602  seq3f1olemstep  10659  seq3f1olemp  10660  nfsum1  11667  nfsum  11668  fsumrelem  11782  nfcprod1  11865  nfcprod  11866  ctiunctlemfo  12810  ctiunct  12811  prdsbas3  13119  cnmpt11  14755  cnmpt21  14763  lgseisenlem2  15548