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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibvalrel | Structured version Visualization version GIF version |
Description: The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
dibcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibcl.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibvalrel | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5706 | . . 3 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) | |
2 | dibcl.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2734 | . . . . . . . 8 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
4 | dibcl.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | dibdiadm 41137 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊)) |
6 | 5 | eleq2d 2824 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊))) |
7 | 6 | biimpa 476 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) |
8 | eqid 2734 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
9 | eqid 2734 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
10 | eqid 2734 | . . . . . 6 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
11 | 8, 2, 9, 10, 3, 4 | dibval 41124 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
12 | 7, 11 | syldan 591 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
13 | 12 | releqd 5790 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (Rel (𝐼‘𝑋) ↔ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))) |
14 | 1, 13 | mpbiri 258 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋)) |
15 | rel0 5811 | . . . 4 ⊢ Rel ∅ | |
16 | ndmfv 6941 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅) | |
17 | 16 | releqd 5790 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (Rel (𝐼‘𝑋) ↔ Rel ∅)) |
18 | 15, 17 | mpbiri 258 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋)) |
19 | 18 | adantl 481 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ ¬ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋)) |
20 | 14, 19 | pm2.61dan 813 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∅c0 4338 {csn 4630 ↦ cmpt 5230 I cid 5581 × cxp 5686 dom cdm 5688 ↾ cres 5690 Rel wrel 5693 ‘cfv 6562 Basecbs 17244 LHypclh 39966 LTrncltrn 40083 DIsoAcdia 41010 DIsoBcdib 41120 |
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-dib 41121 |
This theorem is referenced by: dibglbN 41148 dib2dim 41225 dih2dimbALTN 41227 |
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