Theorem nffo 5546

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 5323	. 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 5416	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 4968	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nffo.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfeq 2380	. . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵
8	4, 7	nfan 1611	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
9	1, 8	nfxfr 1520	1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵

Syntax hints:   ∧ wa 104   = wceq 1395  Ⅎwnf 1506  Ⅎwnfc 2359  ran crn 4719   Fn wfn 5312  –onto→wfo 5315

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-fo 5323

This theorem is referenced by:  nff1o  5569  ctiunctal  13007