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Theorem foconst 6085
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 6009 . . . . 5 (𝐹:𝐴⟶{𝐵} → Rel 𝐹)
2 relrn0 5347 . . . . . 6 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
32necon3abid 2832 . . . . 5 (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
41, 3syl 17 . . . 4 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
5 frn 6012 . . . . . 6 (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵})
6 sssn 4331 . . . . . 6 (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
75, 6sylib 208 . . . . 5 (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
87ord 392 . . . 4 (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵}))
94, 8sylbid 230 . . 3 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵}))
109imdistani 725 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
11 dffo2 6078 . 2 (𝐹:𝐴onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
1210, 11sylibr 224 1 ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴onto→{𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wne 2796  wss 3560  c0 3896  {csn 4153  ran crn 5080  Rel wrel 5084  wf 5846  ontowfo 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-fun 5852  df-fn 5853  df-f 5854  df-fo 5856
This theorem is referenced by: (None)
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