Theorem nff 6687

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 6518	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		nff.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nff.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffn 6620	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5919	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nff.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfss 3942	. . 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 2877   ⊆ wss 3917  ran crn 5642   Fn wfn 6509  ⟶wf 6510

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-f 6518

This theorem is referenced by:  nff1  6757  dffo3f  7081  nfwrd  14515  lfgrnloop  29059  fcomptf  32589  aciunf1lem  32593  fnpreimac  32602  esumfzf  34066  esumfsup  34067  poimirlem24  37645  sdclem1  37744  nfrelp  44946  fmuldfeqlem1  45587  fnlimfvre  45679  dvnmul  45948  stoweidlem53  46058  stoweidlem54  46059  stoweidlem57  46062  sge0iunmpt  46423