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Mathbox for Norm Megill |
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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 41165 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) |
6 | 5 | fndmd 6673 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 class class class wbr 5147 dom cdm 5688 ‘cfv 6562 lecple 17304 Atomscatm 39244 LHypclh 39966 DIsoCcdic 41154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-dic 41155 |
This theorem is referenced by: dicvalrelN 41167 |
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