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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibfna | Structured version Visualization version GIF version |
Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) |
Ref | Expression |
---|---|
dibfna.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibfna.j | ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) |
dibfna.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibfna | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6424 | . . . 4 ⊢ (𝐽‘𝑦) ∈ V | |
2 | snex 5099 | . . . 4 ⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V | |
3 | 1, 2 | xpex 7196 | . . 3 ⊢ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V |
4 | eqid 2799 | . . 3 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) | |
5 | 3, 4 | fnmpti 6233 | . 2 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽 |
6 | eqid 2799 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
7 | dibfna.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | eqid 2799 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
9 | eqid 2799 | . . . 4 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
10 | dibfna.j | . . . 4 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) | |
11 | dibfna.i | . . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
12 | 6, 7, 8, 9, 10, 11 | dibfval 37162 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))) |
13 | 12 | fneq1d 6192 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn dom 𝐽 ↔ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽)) |
14 | 5, 13 | mpbiri 250 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {csn 4368 ↦ cmpt 4922 I cid 5219 × cxp 5310 dom cdm 5312 ↾ cres 5314 Fn wfn 6096 ‘cfv 6101 Basecbs 16184 LHypclh 36005 LTrncltrn 36122 DIsoAcdia 37049 DIsoBcdib 37159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-dib 37160 |
This theorem is referenced by: dibdiadm 37176 dibfnN 37177 dibclN 37183 |
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