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Theorem nffo 5030
 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 4835 . 2 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
2 nffo.1 . . . 4 x𝐹
3 nffo.2 . . . 4 xA
42, 3nffn 4921 . . 3 x 𝐹 Fn A
52nfrn 4506 . . . 4 xran 𝐹
6 nffo.3 . . . 4 xB
75, 6nfeq 2167 . . 3 xran 𝐹 = B
84, 7nfan 1439 . 2 x(𝐹 Fn A ran 𝐹 = B)
91, 8nfxfr 1343 1 x 𝐹:AontoB
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228  Ⅎwnf 1329  Ⅎwnfc 2147  ran crn 4273   Fn wfn 4824  –onto→wfo 4827 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-fun 4831  df-fn 4832  df-fo 4835 This theorem is referenced by:  nff1o  5049
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