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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 5703 | . . 3 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) | |
| 2 | dibcl.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | eqid 2737 | . . . . . . . 8 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
| 4 | dibcl.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
| 5 | 2, 3, 4 | dibdiadm 41157 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊)) |
| 6 | 5 | eleq2d 2827 | . . . . . 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 2737 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 10 | eqid 2737 | . . . . . 6 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
| 11 | 8, 2, 9, 10, 3, 4 | dibval 41144 | . . . . 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 5788 | . . 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 5809 | . . . 4 ⊢ Rel ∅ | |
| 16 | ndmfv 6941 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅) | |
| 17 | 16 | releqd 5788 | . . . 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 1540 ∈ wcel 2108 ∅c0 4333 {csn 4626 ↦ cmpt 5225 I cid 5577 × cxp 5683 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 ‘cfv 6561 Basecbs 17247 LHypclh 39986 LTrncltrn 40103 DIsoAcdia 41030 DIsoBcdib 41140 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-dib 41141 |
| This theorem is referenced by: dibglbN 41168 dib2dim 41245 dih2dimbALTN 41247 |
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