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Theorem foconst 6813
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 6715 . . . . 5 (𝐹:𝐴⟶{𝐵} → Rel 𝐹)
2 relrn0 5961 . . . . . 6 (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
32necon3abid 2971 . . . . 5 (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
41, 3syl 17 . . . 4 (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
5 frn 6717 . . . . . 6 (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵})
6 sssn 4824 . . . . . 6 (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
75, 6sylib 217 . . . . 5 (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
87ord 861 . . . 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 6802 . 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 844   = wceq 1533  wne 2934  wss 3943  c0 4317  {csn 4623  ran crn 5670  Rel wrel 5674  wf 6532  ontowfo 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542
This theorem is referenced by:  dif1enlem  9155  dif1enlemOLD  9156  fullthinc  47922
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