Theorem nffo 6564

Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Hypotheses

Ref	Expression
nffo.1	⊢ Ⅎ𝑥𝐹
nffo.2	⊢ Ⅎ𝑥𝐴
nffo.3	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nffo	⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵

Proof of Theorem nffo

Step	Hyp	Ref	Expression
1		df-fo 6330	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
2		nffo.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nffo.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffn 6422	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5788	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nffo.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfeq 2968	. . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵
8	4, 7	nfan 1900	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
9	1, 8	nfxfr 1854	1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵

Syntax hints:   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  Ⅎwnfc 2936  ran crn 5520   Fn wfn 6319  –onto→wfo 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-fun 6326  df-fn 6327  df-fo 6330

This theorem is referenced by:  nff1o  6588  fompt  41819