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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 6597 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → Rel 𝐹) | |
2 | relrn0 5871 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅)) | |
3 | 2 | necon3abid 2980 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
5 | frn 6599 | . . . . . 6 ⊢ (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵}) | |
6 | sssn 4759 | . . . . . 6 ⊢ (ran 𝐹 ⊆ {𝐵} ↔ (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵})) | |
7 | 5, 6 | sylib 217 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → (ran 𝐹 = ∅ ∨ ran 𝐹 = {𝐵})) |
8 | 7 | ord 861 | . . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (¬ ran 𝐹 = ∅ → ran 𝐹 = {𝐵})) |
9 | 4, 8 | sylbid 239 | . . 3 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ → ran 𝐹 = {𝐵})) |
10 | 9 | imdistani 569 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵})) |
11 | dffo2 6684 | . 2 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵})) | |
12 | 10, 11 | sylibr 233 | 1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ≠ wne 2943 ⊆ wss 3886 ∅c0 4256 {csn 4561 ran crn 5585 Rel wrel 5589 ⟶wf 6422 –onto→wfo 6424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5074 df-opab 5136 df-xp 5590 df-rel 5591 df-cnv 5592 df-dm 5594 df-rn 5595 df-fun 6428 df-fn 6429 df-f 6430 df-fo 6432 |
This theorem is referenced by: dif1enlem 8930 fullthinc 46305 |
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