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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dibn0 | Structured version Visualization version GIF version | ||
| Description: The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.) | 
| Ref | Expression | 
|---|---|
| dibn0.b | ⊢ 𝐵 = (Base‘𝐾) | 
| dibn0.l | ⊢ ≤ = (le‘𝐾) | 
| dibn0.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| dibn0.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | 
| Ref | Expression | 
|---|---|
| dibn0 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dibn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | dibn0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 3 | dibn0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 4 | eqid 2736 | . . 3 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 5 | eqid 2736 | . . 3 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) | |
| 6 | eqid 2736 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
| 7 | dibn0.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 41147 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) | 
| 9 | 1, 2, 3, 6 | dian0 41042 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((DIsoA‘𝐾)‘𝑊)‘𝑋) ≠ ∅) | 
| 10 | fvex 6918 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
| 11 | 10 | mptex 7244 | . . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ V | 
| 12 | 11 | snnz 4775 | . . . 4 ⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ≠ ∅ | 
| 13 | 9, 12 | jctir 520 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((((DIsoA‘𝐾)‘𝑊)‘𝑋) ≠ ∅ ∧ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ≠ ∅)) | 
| 14 | xpnz 6178 | . . 3 ⊢ (((((DIsoA‘𝐾)‘𝑊)‘𝑋) ≠ ∅ ∧ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ≠ ∅) ↔ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ≠ ∅) | |
| 15 | 13, 14 | sylib 218 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) ≠ ∅) | 
| 16 | 8, 15 | eqnetrd 3007 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 I cid 5576 × cxp 5682 ↾ cres 5686 ‘cfv 6560 Basecbs 17248 lecple 17305 HLchlt 39352 LHypclh 39987 LTrncltrn 40104 DIsoAcdia 41031 DIsoBcdib 41141 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-disoa 41032 df-dib 41142 | 
| This theorem is referenced by: dibord 41162 diblss 41173 | 
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