Theorem foconst 6756

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 6662	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → Rel 𝐹)
2		relrn0 5917	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
3	2	necon3abid 2966	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
4	1, 3	syl 17	. . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅))
5		frn 6664	. . . . . 6 ⊢ (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵})
6		sssn 4759	. . . . . 6 ⊢ (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
7	5, 6	sylib 218	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵}))
8	7	ord 865	. . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵}))
9	4, 8	sylbid 240	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵}))
10	9	imdistani 568	. 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
11		dffo2 6745	. 2 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵}))
12	10, 11	sylibr 234	1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ≠ wne 2930   ⊆ wss 3885  ∅c0 4263  {csn 4557  ran crn 5621  Rel wrel 5625  ⟶wf 6483  –onto→wfo 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493

This theorem is referenced by:  dif1enlem  9083  fullthinc  49913