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Mirrors > Home > MPE Home > Th. List > Mathboxes > diaeldm | Structured version Visualization version GIF version |
Description: Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.) |
Ref | Expression |
---|---|
diafn.b | ⊢ 𝐵 = (Base‘𝐾) |
diafn.l | ⊢ ≤ = (le‘𝐾) |
diafn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
diafn.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
diaeldm | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diafn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | diafn.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | diafn.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | diafn.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | diadm 38165 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
6 | 5 | eleq2d 2898 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})) |
7 | breq1 5062 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊)) | |
8 | 7 | elrab 3680 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) |
9 | 6, 8 | syl6bb 289 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5059 dom cdm 5550 ‘cfv 6350 Basecbs 16477 lecple 16566 LHypclh 37114 DIsoAcdia 38158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-disoa 38159 |
This theorem is referenced by: diadmclN 38167 diadmleN 38168 dia0eldmN 38170 dia1eldmN 38171 diaf11N 38179 diaglbN 38185 diaintclN 38188 diasslssN 38189 docaclN 38254 doca2N 38256 djajN 38267 dibval2 38274 dibeldmN 38288 |
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