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 5841 | . . . . . 6 ⊢ Fun 𝐼 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → Fun 𝐼) |
5 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → Fun 𝐼) |
6 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
7 | 2 | dmeqi 5247 | . . . . . . . 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 4859 | . . . . . . . . . 10 ⊢ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V | |
10 | 9 | 2a1i 12 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V)) |
11 | 1, 10 | ralrimi 2940 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
12 | dmmptg 5549 | . . . . . . . 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 2645 | . . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼) |
15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼) |
16 | 6, 15 | eleqtrd 2690 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼) |
17 | fvco 6184 | . . . 4 ⊢ ((Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) | |
18 | 5, 16, 17 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) |
19 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
20 | 2 | fvmpt2 6200 | . . . . 5 ⊢ ((𝑘 ∈ 𝑋 ∧ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
21 | 6, 19, 20 | syl2anc 691 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
22 | 21 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉)) |
23 | df-ov 6552 | . . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉) | |
24 | 23 | eqcomi 2619 | . . . 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 2648 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
27 | 1, 26 | ixpeq2d 38262 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 dom cdm 5038 ∘ ccom 5042 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ov 6552 df-ixp 7795 |
This theorem is referenced by: opnvonmbllem1 39522 |
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