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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibf11N | Structured version Visualization version GIF version |
Description: The partial isomorphism A for a lattice 𝐾 is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibcl.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibf11N | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2610 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | dibcl.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dibcl.i | . . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | dibfnN 35463 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊}) |
6 | fnfun 5902 | . . . 4 ⊢ (𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} → Fun 𝐼) | |
7 | funfn 5833 | . . . 4 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
8 | 6, 7 | sylib 207 | . . 3 ⊢ (𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} → 𝐼 Fn dom 𝐼) |
9 | 5, 8 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐼) |
10 | eqidd 2611 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 = ran 𝐼) | |
11 | 1, 2, 3, 4 | dibeldmN 35465 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ dom 𝐼 ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))) |
12 | 1, 2, 3, 4 | dibeldmN 35465 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊))) |
13 | 11, 12 | anbi12d 743 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ dom 𝐼 ∧ 𝑦 ∈ dom 𝐼) ↔ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))) |
14 | 1, 2, 3, 4 | dib11N 35467 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼‘𝑥) = (𝐼‘𝑦) ↔ 𝑥 = 𝑦)) |
15 | 14 | biimpd 218 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦)) |
16 | 15 | 3expib 1260 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦))) |
17 | 13, 16 | sylbid 229 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ dom 𝐼 ∧ 𝑦 ∈ dom 𝐼) → ((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦))) |
18 | 17 | ralrimivv 2953 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦)) |
19 | dff1o6 6431 | . 2 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 ↔ (𝐼 Fn dom 𝐼 ∧ ran 𝐼 = ran 𝐼 ∧ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦))) | |
20 | 9, 10, 18, 19 | syl3anbrc 1239 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 class class class wbr 4583 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 –1-1-onto→wf1o 5803 ‘cfv 5804 Basecbs 15695 lecple 15775 HLchlt 33655 LHypclh 34288 DIsoBcdib 35445 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-undef 7286 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-disoa 35336 df-dib 35446 |
This theorem is referenced by: dibintclN 35474 |
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