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Theorem nffo 6027
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 5810 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
2 nffo.1 . . . 4 𝑥𝐹
3 nffo.2 . . . 4 𝑥𝐴
42, 3nffn 5901 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5289 . . . 4 𝑥ran 𝐹
6 nffo.3 . . . 4 𝑥𝐵
75, 6nfeq 2762 . . 3 𝑥ran 𝐹 = 𝐵
84, 7nfan 1816 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
91, 8nfxfr 1771 1 𝑥 𝐹:𝐴onto𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wnf 1699  wnfc 2738  ran crn 5039   Fn wfn 5799  ontowfo 5802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-fo 5810
This theorem is referenced by:  nff1o  6048  fompt  38374
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