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Mirrors > Home > MPE Home > Th. List > Mathboxes > dicdmN | Structured version Visualization version GIF version |
Description: Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dicfn.l | ⊢ ≤ = (le‘𝐾) |
dicfn.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dicfn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dicfn.i | ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dicdmN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dicfn.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | dicfn.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | dicfn.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dicfn.i | . . 3 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | dicfnN 36472 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) |
6 | fndm 5990 | . 2 ⊢ (𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} → dom 𝐼 = {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) | |
7 | 5, 6 | syl 17 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 class class class wbr 4653 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 lecple 15948 Atomscatm 34550 LHypclh 35270 DIsoCcdic 36461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-dic 36462 |
This theorem is referenced by: dicvalrelN 36474 |
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