Theorem nff 6648

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	⊢ Ⅎ𝑥𝐹
nff.2	⊢ Ⅎ𝑥𝐴
nff.3	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nff	⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵

Proof of Theorem nff

Step	Hyp	Ref	Expression
1		df-f 6486	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		nff.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nff.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffn 6581	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5894	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nff.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfss 3928	. . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵
8	4, 7	nfan 1899	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)
9	1, 8	nfxfr 1853	1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵

Syntax hints:   ∧ wa 395  Ⅎwnf 1783  Ⅎwnfc 2876   ⊆ wss 3903  ran crn 5620   Fn wfn 6477  ⟶wf 6478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-fun 6484  df-fn 6485  df-f 6486

This theorem is referenced by:  nff1  6718  dffo3f  7040  nfwrd  14450  lfgrnloop  29070  fcomptf  32601  aciunf1lem  32605  fnpreimac  32614  esumfzf  34036  esumfsup  34037  poimirlem24  37628  sdclem1  37727  nfrelp  44927  fmuldfeqlem1  45567  fnlimfvre  45659  dvnmul  45928  stoweidlem53  46038  stoweidlem54  46039  stoweidlem57  46042  sge0iunmpt  46403