Theorem hoi2toco 47362

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 6582	. . . . . 6 ⊢ Fun 𝐼
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → Fun 𝐼)
5	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → Fun 𝐼)
6		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
7	2	dmeqi 5899	. . . . . . . 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 5450	. . . . . . . . . 10 ⊢ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V
10	9	2a1i 12	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V))
11	1, 10	ralrimi 3266	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
12		dmmptg 6248	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝑋 ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V → dom (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) = 𝑋)
13	11, 12	syl 18	. . . . . . 7 ⊢ (𝜑 → dom (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) = 𝑋)
14	8, 13	eqtr2d 2802	. . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼)
15	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼)
16	6, 15	eleqtrd 2868	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼)
17		fvco 6986	. . . 4 ⊢ ((Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘)))
18	5, 16, 17	syl2anc 596	. . 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 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
22	21	fveq2d 6892	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
23		df-ov 7426	. . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
24	23	eqcomi 2775	. . . 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 2805	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
27	1, 26	ixpeq2d 45829	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3082  Vcvv 3458  ⟨cop 4600   ↦ cmpt 5197  dom cdm 5666   ∘ ccom 5670  Fun wfun 6537  ‘cfv 6543  (class class class)co 7423  Xcixp 8904  [,)cico 13392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7426  df-ixp 8905

This theorem is used by:  opnvonmbllem1  47387