Theorem hoicoto2 45619

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 479	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
4	2, 3	fvovco 44190	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))))
5	1	ffvelcdmda 7085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 8009	. . . . . . . 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 3493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ V)
9		hoicoto2.a	. . . . . . 7 ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘)))
10	9	fvmpt2 7008	. . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (1st ‘(𝐼‘𝑘)) ∈ V) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘)))
11	3, 8, 10	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘)))
12	11	eqcomd 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) = (𝐴‘𝑘))
13		xp2nd 8010	. . . . . . . 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 3493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ V)
16		hoicoto2.b	. . . . . . 7 ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘)))
17	16	fvmpt2 7008	. . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (2nd ‘(𝐼‘𝑘)) ∈ V) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘)))
18	3, 15, 17	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘)))
19	18	eqcomd 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) = (𝐵‘𝑘))
20	12, 19	oveq12d 7429	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21	4, 20	eqtrd 2770	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
22	21	ixpeq2dva 8908	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ↦ cmpt 5230   × cxp 5673   ∘ ccom 5679  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411  1^st c1st 7975  2^nd c2nd 7976  Xcixp 8893  ℝcr 11111  [,)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  ax-un 7727

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-f 6546  df-fv 6550  df-ov 7414  df-1st 7977  df-2nd 7978  df-ixp 8894

This theorem is referenced by:  opnvonmbllem2  45647