Theorem hoicoto2 47658

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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
4	2, 3	fvovco 46251	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))))
5	1	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 8033	. . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼‘𝑘)) ∈ ℝ)
7	5, 6	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ ℝ)
8	7	elexd 3473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ V)
9		hoicoto2.a	. . . . . . 7 ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘)))
10	9	fvmpt2 7005	. . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (1st ‘(𝐼‘𝑘)) ∈ V) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘)))
11	3, 8, 10	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘)))
12	11	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) = (𝐴‘𝑘))
13		xp2nd 8034	. . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ)
14	5, 13	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ)
15	14	elexd 3473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ V)
16		hoicoto2.b	. . . . . . 7 ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘)))
17	16	fvmpt2 7005	. . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (2nd ‘(𝐼‘𝑘)) ∈ V) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘)))
18	3, 15, 17	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘)))
19	18	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) = (𝐵‘𝑘))
20	12, 19	oveq12d 7438	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21	4, 20	eqtrd 2795	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
22	21	ixpeq2dva 8940	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185   × cxp 5648   ∘ ccom 5654  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Xcixp 8925  ℝcr 11199  [,)cico 13478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-1st 8001  df-2nd 8002  df-ixp 8926

This theorem is used by:  opnvonmbllem2  47686