Theorem nff 6505

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 6354	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		nff.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nff.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffn 6447	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5819	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nff.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfss 3960	. . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵
8	4, 7	nfan 1896	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)
9	1, 8	nfxfr 1849	1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵

Syntax hints:   ∧ wa 398  Ⅎwnf 1780  Ⅎwnfc 2961   ⊆ wss 3936  ran crn 5551   Fn wfn 6345  ⟶wf 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-fun 6352  df-fn 6353  df-f 6354

This theorem is referenced by:  nff1  6568  nfwrd  13888  lfgrnloop  26904  fcomptf  30397  aciunf1lem  30401  fnpreimac  30410  esumfzf  31323  esumfsup  31324  poimirlem24  34910  sdclem1  35012  dffo3f  41430  fmuldfeqlem1  41855  fnlimfvre  41947  dvnmul  42220  stoweidlem53  42331  stoweidlem54  42332  stoweidlem57  42335  sge0iunmpt  42693