Theorem nff 5222

Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nff.1	|- F/_xF
nff.2	|- F/_xA
nff.3	|- F/_xB

Assertion

Ref	Expression
nff	|- F/x F:A-->B

Proof of Theorem nff

Step	Hyp	Ref	Expression
1		df-f 4792	. 2 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
2		nff.1	. . . 4 |- F/_xF
3		nff.2	. . . 4 |- F/_xA
4	2, 3	nffn 5181	. . 3 |- F/x F Fn A
5	2	nfrn 4964	. . . 4 |- F/_xran F
6		nff.3	. . . 4 |- F/_xB
7	5, 6	nfss 3267	. . 3 |- F/xran F C_ B
8	4, 7	nfan 1824	. 2 |- F/x(F Fn A /\ ran F C_ B)
9	1, 8	nfxfr 1570	1 |- F/x F:A-->B

Syntax hints:   /\ wa 358   F/wnf 1544   F/_wnfc 2477   C_ wss 3258  ran crn 4774   Fn wfn 4777  -->wf 4778

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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792

This theorem is referenced by:  nff1  5257