Theorem nffo 6792

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 6543	. 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 6635	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 5943	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nffo.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfeq 2944	. . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵
8	4, 7	nfan 1926	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
9	1, 8	nfxfr 1880	1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵

Syntax hints:   ∧ wa 400   = wceq 1567  Ⅎwnf 1810  Ⅎwnfc 2916  ran crn 5663   Fn wfn 6532  –onto→wfo 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-fun 6539  df-fn 6540  df-fo 6543

This theorem is referenced by:  nff1o  6819  fompt  7114