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Theorem foconst 6826
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 6727 . . . . 5 (𝐹:𝐴⟶{𝐵} → Rel 𝐹)
2 relrn0 5972 . . . . . 6 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
32necon3abid 2974 . . . . 5 (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
41, 3syl 17 . . . 4 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
5 frn 6729 . . . . . 6 (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵})
6 sssn 4830 . . . . . 6 (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
75, 6sylib 217 . . . . 5 (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
87ord 863 . . . 4 (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵}))
94, 8sylbid 239 . . 3 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵}))
109imdistani 568 . 2 ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
11 dffo2 6815 . 2 (𝐹:𝐴onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
1210, 11sylibr 233 1 ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴onto→{𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wne 2937  wss 3947  c0 4323  {csn 4629  ran crn 5679  Rel wrel 5683  wf 6544  ontowfo 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554
This theorem is referenced by:  dif1enlem  9181  dif1enlemOLD  9182  fullthinc  48052
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