Theorem nff 6666

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 6503	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		nff.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nff.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffn 6599	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5905	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nff.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfss 3936	. . 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 3911  ran crn 5632   Fn wfn 6494  ⟶wf 6495

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6501  df-fn 6502  df-f 6503

This theorem is referenced by:  nff1  6736  dffo3f  7060  nfwrd  14484  lfgrnloop  29028  fcomptf  32555  aciunf1lem  32559  fnpreimac  32568  esumfzf  34032  esumfsup  34033  poimirlem24  37611  sdclem1  37710  nfrelp  44912  fmuldfeqlem1  45553  fnlimfvre  45645  dvnmul  45914  stoweidlem53  46024  stoweidlem54  46025  stoweidlem57  46028  sge0iunmpt  46389