Theorem hoicoto2 44143

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
hoicoto2.i	⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))
hoicoto2.a	⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘)))
hoicoto2.b	⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘)))

Assertion

Ref	Expression
hoicoto2	⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

Distinct variable groups:   𝑘,𝑋   𝜑,𝑘

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

Proof of Theorem hoicoto2

Step	Hyp	Ref	Expression
1		hoicoto2.i	. . . . 5 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))
2	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
4	2, 3	fvovco 42732	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))))
5	1	ffvelrnda 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 7863	. . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼‘𝑘)) ∈ ℝ)
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ ℝ)
8	7	elexd 3452	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ V)
9		hoicoto2.a	. . . . . . 7 ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘)))
10	9	fvmpt2 6886	. . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (1st ‘(𝐼‘𝑘)) ∈ V) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘)))
11	3, 8, 10	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘)))
12	11	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) = (𝐴‘𝑘))
13		xp2nd 7864	. . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ)
14	5, 13	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ)
15	14	elexd 3452	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ V)
16		hoicoto2.b	. . . . . . 7 ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘)))
17	16	fvmpt2 6886	. . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (2nd ‘(𝐼‘𝑘)) ∈ V) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘)))
18	3, 15, 17	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘)))
19	18	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) = (𝐵‘𝑘))
20	12, 19	oveq12d 7293	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21	4, 20	eqtrd 2778	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
22	21	ixpeq2dva 8700	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ↦ cmpt 5157   × cxp 5587   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Xcixp 8685  ℝcr 10870  [,)cico 13081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-1st 7831  df-2nd 7832  df-ixp 8686

This theorem is referenced by:  opnvonmbllem2  44171