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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 6732 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → Rel 𝐹) | |
2 | relrn0 5976 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅)) | |
3 | 2 | necon3abid 2974 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
5 | frn 6734 | . . . . . 6 ⊢ (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵}) | |
6 | sssn 4834 | . . . . . 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 862 | . . . 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 567 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵})) |
11 | dffo2 6820 | . 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 394 ∨ wo 845 = wceq 1533 ≠ wne 2937 ⊆ wss 3949 ∅c0 4326 {csn 4632 ran crn 5683 Rel wrel 5687 ⟶wf 6549 –onto→wfo 6551 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 |
This theorem is referenced by: dif1enlem 9187 dif1enlemOLD 9188 fullthinc 48130 |
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