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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dibopelvalN | Structured version Visualization version GIF version | ||
| Description: Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dibval.b | ⊢ 𝐵 = (Base‘𝐾) |
| dibval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| dibval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| dibval.o | ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
| dibval.j | ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) |
| dibval.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
| Ref | Expression |
|---|---|
| dibopelvalN | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dibval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | dibval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | dibval.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 4 | dibval.o | . . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
| 5 | dibval.j | . . . 4 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) | |
| 6 | dibval.i | . . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
| 7 | 1, 2, 3, 4, 5, 6 | dibval 41144 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
| 8 | 7 | eleq2d 2827 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ 〈𝐹, 𝑆〉 ∈ ((𝐽‘𝑋) × { 0 }))) |
| 9 | opelxp 5721 | . . 3 ⊢ (〈𝐹, 𝑆〉 ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 })) | |
| 10 | 3 | fvexi 6920 | . . . . . . 7 ⊢ 𝑇 ∈ V |
| 11 | 10 | mptex 7243 | . . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V |
| 12 | 4, 11 | eqeltri 2837 | . . . . 5 ⊢ 0 ∈ V |
| 13 | 12 | elsn2 4665 | . . . 4 ⊢ (𝑆 ∈ { 0 } ↔ 𝑆 = 0 ) |
| 14 | 13 | anbi2i 623 | . . 3 ⊢ ((𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )) |
| 15 | 9, 14 | bitri 275 | . 2 ⊢ (〈𝐹, 𝑆〉 ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )) |
| 16 | 8, 15 | bitrdi 287 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ↦ cmpt 5225 I cid 5577 × cxp 5683 dom cdm 5685 ↾ cres 5687 ‘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: (None) |
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