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Theorem foconst 6039
Description: A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
Assertion
Ref Expression
foconst ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴onto→{𝐵})

Proof of Theorem foconst
StepHypRef Expression
1 frel 5963 . . . . 5 (𝐹:𝐴⟶{𝐵} → Rel 𝐹)
2 relrn0 5304 . . . . . 6 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
32necon3abid 2818 . . . . 5 (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
41, 3syl 17 . . . 4 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
5 frn 5966 . . . . . 6 (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵})
6 sssn 4298 . . . . . 6 (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
75, 6sylib 207 . . . . 5 (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
87ord 391 . . . 4 (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵}))
94, 8sylbid 229 . . 3 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵}))
109imdistani 722 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
11 dffo2 6032 . 2 (𝐹:𝐴onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
1210, 11sylibr 223 1 ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴onto→{𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wne 2780  wss 3540  c0 3874  {csn 4125  ran crn 5039  Rel wrel 5043  wf 5800  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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  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-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810
This theorem is referenced by: (None)
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