Theorem hoi2toco 44534

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 6524	. . . . . 6 ⊢ Fun 𝐼
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → Fun 𝐼)
5	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → Fun 𝐼)
6		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
7	2	dmeqi 5847	. . . . . . . 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 5410	. . . . . . . . . 10 ⊢ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V
10	9	2a1i 12	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V))
11	1, 10	ralrimi 3236	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
12		dmmptg 6181	. . . . . . . 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 2777	. . . . . 6 ⊢ (𝜑 → 𝑋 = dom 𝐼)
15	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 = dom 𝐼)
16	6, 15	eleqtrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ dom 𝐼)
17		fvco 6923	. . . 4 ⊢ ((Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘)))
18	5, 16, 17	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼‘𝑘)))
19	9	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
20	2	fvmpt2 6943	. . . . 5 ⊢ ((𝑘 ∈ 𝑋 ∧ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V) → (𝐼‘𝑘) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
21	6, 19, 20	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
22	21	fveq2d 6830	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ([,)‘(𝐼‘𝑘)) = ([,)‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
23		df-ov 7341	. . . . 5 ⊢ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ([,)‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
24	23	eqcomi 2745	. . . 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 2780	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
27	1, 26	ixpeq2d 42988	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  ∀wral 3061  Vcvv 3441  ⟨cop 4580   ↦ cmpt 5176  dom cdm 5621   ∘ ccom 5625  Fun wfun 6474  ‘cfv 6480  (class class class)co 7338  Xcixp 8757  [,)cico 13183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-fv 6488  df-ov 7341  df-ixp 8758

This theorem is referenced by:  opnvonmbllem1  44559