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Theorem nffo 6580
 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 6349 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
2 nffo.1 . . . 4 𝑥𝐹
3 nffo.2 . . . 4 𝑥𝐴
42, 3nffn 6440 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5811 . . . 4 𝑥ran 𝐹
6 nffo.3 . . . 4 𝑥𝐵
75, 6nfeq 2995 . . 3 𝑥ran 𝐹 = 𝐵
84, 7nfan 1901 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
91, 8nfxfr 1854 1 𝑥 𝐹:𝐴onto𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  Ⅎwnfc 2962  ran crn 5543   Fn wfn 6338  –onto→wfo 6341 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-fun 6345  df-fn 6346  df-fo 6349 This theorem is referenced by:  nff1o  6604  fompt  41744
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