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 6387 | . . . . . 6 ⊢ Fun 𝐼 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → Fun 𝐼) |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → Fun 𝐼) |
6 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
7 | 2 | dmeqi 5766 | . . . . . . . 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 5347 | . . . . . . . . . 10 ⊢ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V | |
10 | 9 | 2a1i 12 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V)) |
11 | 1, 10 | ralrimi 3214 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
12 | dmmptg 6089 | . . . . . . . 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 2855 | . . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼) |
15 | 14 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼) |
16 | 6, 15 | eleqtrd 2913 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼) |
17 | fvco 6752 | . . . 4 ⊢ ((Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) | |
18 | 5, 16, 17 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘))) |
19 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) |
20 | 2 | fvmpt2 6772 | . . . . 5 ⊢ ((𝑘 ∈ 𝑋 ∧ 〈(𝐴‘𝑘), (𝐵‘𝑘)〉 ∈ V) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
21 | 6, 19, 20 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) = 〈(𝐴‘𝑘), (𝐵‘𝑘)〉) |
22 | 21 | fveq2d 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉)) |
23 | df-ov 7151 | . . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘〈(𝐴‘𝑘), (𝐵‘𝑘)〉) | |
24 | 23 | eqcomi 2828 | . . . 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 2858 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
27 | 1, 26 | ixpeq2d 41321 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 Ⅎwnf 1778 ∈ wcel 2108 ∀wral 3136 Vcvv 3493 〈cop 4565 ↦ cmpt 5137 dom cdm 5548 ∘ ccom 5552 Fun wfun 6342 ‘cfv 6348 (class class class)co 7148 Xcixp 8453 [,)cico 12732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-ov 7151 df-ixp 8454 |
This theorem is referenced by: opnvonmbllem1 42905 |
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