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Mirrors > Home > MPE Home > Th. List > Mathboxes > diafn | Structured version Visualization version GIF version |
Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.) |
Ref | Expression |
---|---|
diafn.b | ⊢ 𝐵 = (Base‘𝐾) |
diafn.l | ⊢ ≤ = (le‘𝐾) |
diafn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
diafn.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
diafn | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
2 | 1 | rabex 5199 | . . 3 ⊢ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦} ∈ V |
3 | eqid 2798 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) | |
4 | 2, 3 | fnmpti 6463 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} |
5 | diafn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
6 | diafn.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
7 | diafn.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | eqid 2798 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
9 | eqid 2798 | . . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
10 | diafn.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
11 | 5, 6, 7, 8, 9, 10 | diafval 38327 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦})) |
12 | 11 | fneq1d 6416 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↔ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})) |
13 | 4, 12 | mpbiri 261 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ↦ cmpt 5110 Fn wfn 6319 ‘cfv 6324 Basecbs 16475 lecple 16564 LHypclh 37280 LTrncltrn 37397 trLctrl 37454 DIsoAcdia 38324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-disoa 38325 |
This theorem is referenced by: diadm 38331 diaelrnN 38341 diaf11N 38345 |
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