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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 6363 | . . . . . 6 ⊢ Fun 𝐼 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → Fun 𝐼) |
5 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → Fun 𝐼) |
6 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
7 | 2 | dmeqi 5737 | . . . . . . . 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 5321 | . . . . . . . . . 10 ⊢ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V | |
10 | 9 | 2a1i 12 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V)) |
11 | 1, 10 | ralrimi 3180 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
12 | dmmptg 6063 | . . . . . . . 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 2834 | . . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼) |
15 | 14 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼) |
16 | 6, 15 | eleqtrd 2892 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼) |
17 | fvco 6736 | . . . 4 ⊢ ((Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) | |
18 | 5, 16, 17 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) |
19 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
20 | 2 | fvmpt2 6756 | . . . . 5 ⊢ ((𝑘 ∈ 𝑋 ∧ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
21 | 6, 19, 20 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
22 | 21 | fveq2d 6649 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉)) |
23 | df-ov 7138 | . . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉) | |
24 | 23 | eqcomi 2807 | . . . 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 2837 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
27 | 1, 26 | ixpeq2d 41702 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 〈cop 4531 ↦ cmpt 5110 dom cdm 5519 ∘ ccom 5523 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 Xcixp 8444 [,)cico 12728 |
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-sep 5167 ax-nul 5174 ax-pow 5231 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-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-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-fv 6332 df-ov 7138 df-ixp 8445 |
This theorem is referenced by: opnvonmbllem1 43271 |
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