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Theorem nffo 5356
 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 5141 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
2 nffo.1 . . . 4 𝑥𝐹
3 nffo.2 . . . 4 𝑥𝐴
42, 3nffn 5231 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 4796 . . . 4 𝑥ran 𝐹
6 nffo.3 . . . 4 𝑥𝐵
75, 6nfeq 2291 . . 3 𝑥ran 𝐹 = 𝐵
84, 7nfan 1541 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)
91, 8nfxfr 1451 1 𝑥 𝐹:𝐴onto𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332  Ⅎwnf 1437  Ⅎwnfc 2270  ran crn 4552   Fn wfn 5130  –onto→wfo 5133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-fun 5137  df-fn 5138  df-fo 5141 This theorem is referenced by:  nff1o  5377  ctiunctal  12026
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