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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 5566 | . . 3 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) | |
2 | dibcl.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2818 | . . . . . . . 8 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
4 | dibcl.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | dibdiadm 38171 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊)) |
6 | 5 | eleq2d 2895 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊))) |
7 | 6 | biimpa 477 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) |
8 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
9 | eqid 2818 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
10 | eqid 2818 | . . . . . 6 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
11 | 8, 2, 9, 10, 3, 4 | dibval 38158 | . . . . 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 5646 | . . 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 259 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋)) |
15 | rel0 5665 | . . . 4 ⊢ Rel ∅ | |
16 | ndmfv 6693 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅) | |
17 | 16 | releqd 5646 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (Rel (𝐼‘𝑋) ↔ Rel ∅)) |
18 | 15, 17 | mpbiri 259 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋)) |
19 | 18 | adantl 482 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ ¬ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋)) |
20 | 14, 19 | pm2.61dan 809 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 ↦ cmpt 5137 I cid 5452 × cxp 5546 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 ‘cfv 6348 Basecbs 16471 LHypclh 37000 LTrncltrn 37117 DIsoAcdia 38044 DIsoBcdib 38154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-dib 38155 |
This theorem is referenced by: dibglbN 38182 dib2dim 38259 dih2dimbALTN 38261 |
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