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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 6510 | . . . 4 ⊢ (𝐽‘𝑦) ∈ V | |
2 | snex 5185 | . . . 4 ⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V | |
3 | 1, 2 | xpex 7292 | . . 3 ⊢ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V |
4 | eqid 2773 | . . 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 6319 | . 2 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽 |
6 | eqid 2773 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
7 | dibfna.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | eqid 2773 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
9 | eqid 2773 | . . . 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 37755 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))) |
13 | 12 | fneq1d 6277 | . 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 387 = wceq 1508 ∈ wcel 2051 {csn 4436 ↦ cmpt 5005 I cid 5308 × cxp 5402 dom cdm 5404 ↾ cres 5406 Fn wfn 6181 ‘cfv 6186 Basecbs 16338 LHypclh 36598 LTrncltrn 36715 DIsoAcdia 37642 DIsoBcdib 37752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-dib 37753 |
This theorem is referenced by: dibdiadm 37769 dibfnN 37770 dibclN 37776 |
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