Theorem hoi2toco 45621

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.)

Hypotheses

Ref	Expression
hoi2toco.1	⊢ Ⅎ𝑘𝜑
hoi2toco.c	⊢ 𝐼 = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)

Assertion

Ref	Expression
hoi2toco	⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

Distinct variable group:   𝑘,𝑋

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐵(𝑘)   𝐼(𝑘)

Proof of Theorem hoi2toco

Step	Hyp	Ref	Expression
1		hoi2toco.1	. 2 ⊢ Ⅎ𝑘𝜑
2		hoi2toco.c	. . . . . . 7 ⊢ 𝐼 = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
3	2	funmpt2 6586	. . . . . 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 5903	. . . . . . . 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 5463	. . . . . . . . . 10 ⊢ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V
10	9	2a1i 12	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V))
11	1, 10	ralrimi 3252	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
12		dmmptg 6240	. . . . . . . 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 2771	. . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼)
15	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼)
16	6, 15	eleqtrd 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼)
17		fvco 6988	. . . 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 7008	. . . . 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 6894	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
23		df-ov 7414	. . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
24	23	eqcomi 2739	. . . 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 2774	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
27	1, 26	ixpeq2d 44056	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104  ∀wral 3059  Vcvv 3472  ⟨cop 4633   ↦ cmpt 5230  dom cdm 5675   ∘ ccom 5679  Fun wfun 6536  ‘cfv 6542  (class class class)co 7411  Xcixp 8893  [,)cico 13330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7414  df-ixp 8894

This theorem is referenced by:  opnvonmbllem1  45646