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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 6674 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → Rel 𝐹) | |
2 | relrn0 5925 | . . . . . 6 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅)) | |
3 | 2 | necon3abid 2981 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 ≠ ∅ ↔ ¬ ran 𝐹 = ∅)) |
5 | frn 6676 | . . . . . 6 ⊢ (𝐹:𝐴⟶{𝐵} → ran 𝐹 ⊆ {𝐵}) | |
6 | sssn 4787 | . . . . . 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 863 | . . . 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 570 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ∧ ran 𝐹 = {𝐵})) |
11 | dffo2 6761 | . 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 397 ∨ wo 846 = wceq 1542 ≠ wne 2944 ⊆ wss 3911 ∅c0 4283 {csn 4587 ran crn 5635 Rel wrel 5639 ⟶wf 6493 –onto→wfo 6495 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 |
This theorem is referenced by: dif1enlem 9101 dif1enlemOLD 9102 fullthinc 47073 |
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