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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 2622 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
3 | dibfn.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | dibfna 36443 | . 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 36324 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
8 | 7 | fneq2d 5982 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊) ↔ 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})) |
9 | 4, 8 | mpbid 222 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 class class class wbr 4653 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 Basecbs 15857 lecple 15948 LHypclh 35270 DIsoAcdia 36317 DIsoBcdib 36427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-disoa 36318 df-dib 36428 |
This theorem is referenced by: dibdmN 36446 dibf11N 36450 |
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