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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 5583 | . . 3 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) | |
2 | dibcl.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2738 | . . . . . . . 8 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
4 | dibcl.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | dibdiadm 38932 | . . . . . . 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 480 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) |
8 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
9 | eqid 2738 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
10 | eqid 2738 | . . . . . 6 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
11 | 8, 2, 9, 10, 3, 4 | dibval 38919 | . . . . 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 594 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
13 | 12 | releqd 5664 | . . 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 261 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋)) |
15 | rel0 5683 | . . . 4 ⊢ Rel ∅ | |
16 | ndmfv 6765 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅) | |
17 | 16 | releqd 5664 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (Rel (𝐼‘𝑋) ↔ Rel ∅)) |
18 | 15, 17 | mpbiri 261 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋)) |
19 | 18 | adantl 485 | . 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 399 = wceq 1543 ∈ wcel 2111 ∅c0 4251 {csn 4555 ↦ cmpt 5149 I cid 5468 × cxp 5563 dom cdm 5565 ↾ cres 5567 Rel wrel 5570 ‘cfv 6397 Basecbs 16784 LHypclh 37761 LTrncltrn 37878 DIsoAcdia 38805 DIsoBcdib 38915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-dib 38916 |
This theorem is referenced by: dibglbN 38943 dib2dim 39020 dih2dimbALTN 39022 |
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