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Mirrors > Home > MPE Home > Th. List > foconst | Structured version Visualization version GIF version |
Description: A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.) |
Ref | Expression |
---|---|
foconst | ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frel 5963 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → Rel 𝐹) | |
2 | relrn0 5304 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅)) | |
3 | 2 | necon3abid 2818 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
5 | frn 5966 | . . . . . 6 ⊢ (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵}) | |
6 | sssn 4298 | . . . . . 6 ⊢ (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵})) | |
7 | 5, 6 | sylib 207 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵})) |
8 | 7 | ord 391 | . . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵})) |
9 | 4, 8 | sylbid 229 | . . 3 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵})) |
10 | 9 | imdistani 722 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵})) |
11 | dffo2 6032 | . 2 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵})) | |
12 | 10, 11 | sylibr 223 | 1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 {csn 4125 ran crn 5039 Rel wrel 5043 ⟶wf 5800 –onto→wfo 5802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 |
This theorem is referenced by: (None) |
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