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Mathbox for Norm Megill |
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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 38331 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
6 | 5 | eleq2d 2875 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})) |
7 | breq1 5033 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊)) | |
8 | 7 | elrab 3628 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) |
9 | 6, 8 | syl6bb 290 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 Basecbs 16475 lecple 16564 LHypclh 37280 DIsoAcdia 38324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-disoa 38325 |
This theorem is referenced by: diadmclN 38333 diadmleN 38334 dia0eldmN 38336 dia1eldmN 38337 diaf11N 38345 diaglbN 38351 diaintclN 38354 diasslssN 38355 docaclN 38420 doca2N 38422 djajN 38433 dibval2 38440 dibeldmN 38454 |
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