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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibfnN | Structured version Visualization version GIF version |
Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibfn.b | ⊢ 𝐵 = (Base‘𝐾) |
dibfn.l | ⊢ ≤ = (le‘𝐾) |
dibfn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibfn.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibfnN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibfn.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
3 | dibfn.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | dibfna 38284 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊)) |
5 | dibfn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
6 | dibfn.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
7 | 5, 6, 1, 2 | diadm 38165 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
8 | 7 | fneq2d 6441 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊) ↔ 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})) |
9 | 4, 8 | mpbid 234 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5058 dom cdm 5549 Fn wfn 6344 ‘cfv 6349 Basecbs 16477 lecple 16566 LHypclh 37114 DIsoAcdia 38158 DIsoBcdib 38268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-disoa 38159 df-dib 38269 |
This theorem is referenced by: dibdmN 38287 dibf11N 38291 |
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