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Theorem nffo 6152
 Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nffo.1 𝑥𝐹
nffo.2 𝑥𝐴
nffo.3 𝑥𝐵
Assertion
Ref Expression
nffo 𝑥 𝐹:𝐴onto𝐵

Proof of Theorem nffo
StepHypRef Expression
1 df-fo 5932 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
2 nffo.1 . . . 4 𝑥𝐹
3 nffo.2 . . . 4 𝑥𝐴
42, 3nffn 6025 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5400 . . . 4 𝑥ran 𝐹
6 nffo.3 . . . 4 𝑥𝐵
75, 6nfeq 2805 . . 3 𝑥ran 𝐹 = 𝐵
84, 7nfan 1868 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
91, 8nfxfr 1819 1 𝑥 𝐹:𝐴onto𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523  Ⅎwnf 1748  Ⅎwnfc 2780  ran crn 5144   Fn wfn 5921  –onto→wfo 5924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-fo 5932 This theorem is referenced by:  nff1o  6173  fompt  39693
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