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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoi2toco | Structured version Visualization version GIF version |
Description: The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hoi2toco.1 | ⊢ Ⅎ𝑘𝜑 |
hoi2toco.c | ⊢ 𝐼 = (𝑘 ∈ 𝑋 ↦ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
Ref | Expression |
---|---|
hoi2toco | ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoi2toco.1 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | hoi2toco.c | . . . . . . 7 ⊢ 𝐼 = (𝑘 ∈ 𝑋 ↦ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) | |
3 | 2 | funmpt2 6593 | . . . . . 6 ⊢ Fun 𝐼 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → Fun 𝐼) |
5 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → Fun 𝐼) |
6 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
7 | 2 | dmeqi 5907 | . . . . . . . 8 ⊢ dom 𝐼 = dom (𝑘 ∈ 𝑋 ↦ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → dom 𝐼 = dom (𝑘 ∈ 𝑋 ↦ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉)) |
9 | opex 5466 | . . . . . . . . . 10 ⊢ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V | |
10 | 9 | 2a1i 12 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V)) |
11 | 1, 10 | ralrimi 3244 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
12 | dmmptg 6248 | . . . . . . . 8 ⊢ (∀𝑘 ∈ 𝑋 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V → dom (𝑘 ∈ 𝑋 ↦ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) = 𝑋) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝜑 → dom (𝑘 ∈ 𝑋 ↦ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) = 𝑋) |
14 | 8, 13 | eqtr2d 2766 | . . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼) |
15 | 14 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼) |
16 | 6, 15 | eleqtrd 2827 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼) |
17 | fvco 6995 | . . . 4 ⊢ ((Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) | |
18 | 5, 16, 17 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) |
19 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
20 | 2 | fvmpt2 7015 | . . . . 5 ⊢ ((𝑘 ∈ 𝑋 ∧ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
21 | 6, 19, 20 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
22 | 21 | fveq2d 6900 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉)) |
23 | df-ov 7422 | . . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉) | |
24 | 23 | eqcomi 2734 | . . . 4 ⊢ ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)) |
25 | 24 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
26 | 18, 22, 25 | 3eqtrd 2769 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
27 | 1, 26 | ixpeq2d 44574 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 〈cop 4636 ↦ cmpt 5232 dom cdm 5678 ∘ ccom 5682 Fun wfun 6543 ‘cfv 6549 (class class class)co 7419 Xcixp 8916 [,)cico 13361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-fv 6557 df-ov 7422 df-ixp 8917 |
This theorem is referenced by: opnvonmbllem1 46158 |
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