Theorem nffv 5537

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 5236	. 2 |- ( F ` A ) = ( iota y A F y )
2		nffv.2	. . . 4 |- F/_ x A
3		nffv.1	. . . 4 |- F/_ x F
4		nfcv 2329	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4061	. . 3 |- F/ x A F y
6	5	nfiotaw 5194	. 2 |- F/_ x ( iota y A F y )
7	1, 6	nfcxfr 2326	1 |- F/_ x ( F ` A )

Syntax hints:   F/_wnfc 2316   class class class wbr 4015   iotacio 5188   ` cfv 5228

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236

This theorem is referenced by:  nffvmpt1  5538  nffvd  5539  dffn5imf  5584  fvmptssdm  5613  fvmptf  5621  eqfnfv2f  5630  ralrnmpt  5671  rexrnmpt  5672  ffnfvf  5688  funiunfvdmf  5778  dff13f  5784  nfiso  5820  nfrecs  6322  nffrec  6411  cc2  7280  nfseq  10469  seq3f1olemstep  10515  seq3f1olemp  10516  nfsum1  11378  nfsum  11379  fsumrelem  11493  nfcprod1  11576  nfcprod  11577  ctiunctlemfo  12454  ctiunct  12455  cnmpt11  14136  cnmpt21  14144  lgseisenlem2  14804