Theorem foconst 6811

Description: A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)

Assertion

Ref	Expression
foconst	⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵})

Proof of Theorem foconst

Step	Hyp	Ref	Expression
1		frel 6715	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → Rel 𝐹)
2		relrn0 5954	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
3	2	necon3abid 2991	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
4	1, 3	syl 18	. . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
5		frn 6717	. . . . . 6 ⊢ (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵})
6		sssn 4786	. . . . . 6 ⊢ (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
7	5, 6	sylib 221	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
8	7	ord 878	. . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵}))
9	4, 8	sylbid 243	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵}))
10	9	imdistani 579	. 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
11		dffo2 6800	. 2 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
12	10, 11	sylibr 237	1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ≠ wne 2955   ⊆ wss 3895  ∅c0 4278  {csn 4583  ran crn 5651  Rel wrel 5655  ⟶wf 6534  –onto→wfo 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544

This theorem is used by:  dif1enlem  9175  fullthinc  50601