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Theorem nffo 6751
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 6498 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
2 nffo.1 . . . 4 𝑥𝐹
3 nffo.2 . . . 4 𝑥𝐴
42, 3nffn 6597 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5904 . . . 4 𝑥ran 𝐹
6 nffo.3 . . . 4 𝑥𝐵
75, 6nfeq 2919 . . 3 𝑥ran 𝐹 = 𝐵
84, 7nfan 1903 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
91, 8nfxfr 1856 1 𝑥 𝐹:𝐴onto𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wnf 1786  wnfc 2886  ran crn 5632   Fn wfn 6487  ontowfo 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6494  df-fn 6495  df-fo 6498
This theorem is referenced by:  nff1o  6778  fompt  43137
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