Theorem nffv 5335

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/_xF
nffv.2	|- F/_xA

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 4796	. 2 |- (F` A) = (iotayAFy)
2		nffv.2	. . . 4 |- F/_xA
3		nffv.1	. . . 4 |- F/_xF
4		nfcv 2490	. . . 4 |- F/_xy
5	2, 3, 4	nfbr 4684	. . 3 |- F/x AFy
6	5	nfiota 4344	. 2 |- F/_x(iotayAFy)
7	1, 6	nfcxfr 2487	1 |- F/_x(F` A)

Syntax hints:   F/_wnfc 2477  iotacio 4338   class class class wbr 4640  ` cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796

This theorem is referenced by:  nffvd  5336  eqfnfv2f  5397  ffnfvf  5429  funiunfvf  5469  dff13f  5473  nfiso  5488  fvmptss  5706  fvmptex  5722  fvmptf  5723  fvmptnf  5724