Theorem nffv 5588

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 5280	. 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 4091	. . 3 |- F/ x A F y
6	5	nfiotaw 5237	. 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 4045   iotacio 5231   ` cfv 5272

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 4046  df-iota 5233  df-fv 5280

This theorem is referenced by:  nffvmpt1  5589  nffvd  5590  dffn5imf  5636  fvmptssdm  5666  fvmptf  5674  eqfnfv2f  5683  ralrnmpt  5724  rexrnmpt  5725  ffnfvf  5741  funiunfvdmf  5835  dff13f  5841  nfiso  5877  nfrecs  6395  nffrec  6484  cc2  7381  nfseq  10604  seq3f1olemstep  10661  seq3f1olemp  10662  nfsum1  11700  nfsum  11701  fsumrelem  11815  nfcprod1  11898  nfcprod  11899  ctiunctlemfo  12843  ctiunct  12844  prdsbas3  13152  cnmpt11  14788  cnmpt21  14796  lgseisenlem2  15581