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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 5150 | . . 3 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) | |
2 | dibcl.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2610 | . . . . . . . 8 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
4 | dibcl.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | dibdiadm 35462 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊)) |
6 | 5 | eleq2d 2673 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊))) |
7 | 6 | biimpa 500 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) |
8 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
9 | eqid 2610 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
10 | eqid 2610 | . . . . . 6 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
11 | 8, 2, 9, 10, 3, 4 | dibval 35449 | . . . . 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 486 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
13 | 12 | releqd 5126 | . . 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 247 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋)) |
15 | rel0 5166 | . . . 4 ⊢ Rel ∅ | |
16 | ndmfv 6128 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅) | |
17 | 16 | releqd 5126 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (Rel (𝐼‘𝑋) ↔ Rel ∅)) |
18 | 15, 17 | mpbiri 247 | . . 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 828 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 {csn 4125 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 ‘cfv 5804 Basecbs 15695 LHypclh 34288 LTrncltrn 34405 DIsoAcdia 35335 DIsoBcdib 35445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-dib 35446 |
This theorem is referenced by: dibglbN 35473 dib2dim 35550 dih2dimbALTN 35552 |
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