Theorem nffo 5509

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 5286	. 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 5379	. . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
5	2	nfrn 4932	. . . 4 ⊢ Ⅎ𝑥ran 𝐹
6		nffo.3	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfeq 2357	. . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵
8	4, 7	nfan 1589	. 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
9	1, 8	nfxfr 1498	1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵

Syntax hints:   ∧ wa 104   = wceq 1373  Ⅎwnf 1484  Ⅎwnfc 2336  ran crn 4684   Fn wfn 5275  –onto→wfo 5278

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-fun 5282  df-fn 5283  df-fo 5286

This theorem is referenced by:  nff1o  5532  ctiunctal  12887