Theorem nff 5364

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 5222	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		nff.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nff.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffn 5314	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 4874	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nff.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfss 3150	. . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵
8	4, 7	nfan 1565	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)
9	1, 8	nfxfr 1474	1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵

Syntax hints:   ∧ wa 104  Ⅎwnf 1460  Ⅎwnfc 2306   ⊆ wss 3131  ran crn 4629   Fn wfn 5213  ⟶wf 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222

This theorem is referenced by:  nff1  5421