Theorem nffo 6114

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 5894	. 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 5987	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5368	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nffo.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfeq 2776	. . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵
8	4, 7	nfan 1828	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
9	1, 8	nfxfr 1779	1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵

Syntax hints:   ∧ wa 384   = wceq 1483  Ⅎwnf 1708  Ⅎwnfc 2751  ran crn 5115   Fn wfn 5883  –onto→wfo 5886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-fo 5894

This theorem is referenced by:  nff1o  6135  fompt  39379