Theorem nffv 5527

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 5226	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2319	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4051	. . 3 |- F/ x A F y
6	5	nfiotaw 5184	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2316	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2306   class class class wbr 4005   iotacio 5178   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226

This theorem is referenced by:  nffvmpt1  5528  nffvd  5529  dffn5imf  5573  fvmptssdm  5602  fvmptf  5610  eqfnfv2f  5619  ralrnmpt  5660  rexrnmpt  5661  ffnfvf  5677  funiunfvdmf  5767  dff13f  5773  nfiso  5809  nfrecs  6310  nffrec  6399  cc2  7268  nfseq  10457  seq3f1olemstep  10503  seq3f1olemp  10504  nfsum1  11366  nfsum  11367  fsumrelem  11481  nfcprod1  11564  nfcprod  11565  ctiunctlemfo  12442  ctiunct  12443  cnmpt11  13868  cnmpt21  13876  lgseisenlem2  14536