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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 38186 | . . 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 5069 | . . 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 1537 ∈ wcel 2114 {crab 3142 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 Basecbs 16483 lecple 16572 LHypclh 37135 DIsoAcdia 38179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-disoa 38180 |
This theorem is referenced by: diadmclN 38188 diadmleN 38189 dia0eldmN 38191 dia1eldmN 38192 diaf11N 38200 diaglbN 38206 diaintclN 38209 diasslssN 38210 docaclN 38275 doca2N 38277 djajN 38288 dibval2 38295 dibeldmN 38309 |
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