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Theorem hoicoto2 46131
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 (𝜑𝐼:𝑋⟶(ℝ × ℝ))
hoicoto2.a 𝐴 = (𝑘𝑋 ↦ (1st ‘(𝐼𝑘)))
hoicoto2.b 𝐵 = (𝑘𝑋 ↦ (2nd ‘(𝐼𝑘)))
Assertion
Ref Expression
hoicoto2 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Distinct variable groups:   𝑘,𝑋   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐼(𝑘)

Proof of Theorem hoicoto2
StepHypRef Expression
1 hoicoto2.i . . . . 5 (𝜑𝐼:𝑋⟶(ℝ × ℝ))
21adantr 479 . . . 4 ((𝜑𝑘𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3 simpr 483 . . . 4 ((𝜑𝑘𝑋) → 𝑘𝑋)
42, 3fvovco 44705 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))))
51ffvelcdmda 7093 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐼𝑘) ∈ (ℝ × ℝ))
6 xp1st 8026 . . . . . . . 8 ((𝐼𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼𝑘)) ∈ ℝ)
75, 6syl 17 . . . . . . 7 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) ∈ ℝ)
87elexd 3483 . . . . . 6 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) ∈ V)
9 hoicoto2.a . . . . . . 7 𝐴 = (𝑘𝑋 ↦ (1st ‘(𝐼𝑘)))
109fvmpt2 7015 . . . . . 6 ((𝑘𝑋 ∧ (1st ‘(𝐼𝑘)) ∈ V) → (𝐴𝑘) = (1st ‘(𝐼𝑘)))
113, 8, 10syl2anc 582 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) = (1st ‘(𝐼𝑘)))
1211eqcomd 2731 . . . 4 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) = (𝐴𝑘))
13 xp2nd 8027 . . . . . . . 8 ((𝐼𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼𝑘)) ∈ ℝ)
145, 13syl 17 . . . . . . 7 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) ∈ ℝ)
1514elexd 3483 . . . . . 6 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) ∈ V)
16 hoicoto2.b . . . . . . 7 𝐵 = (𝑘𝑋 ↦ (2nd ‘(𝐼𝑘)))
1716fvmpt2 7015 . . . . . 6 ((𝑘𝑋 ∧ (2nd ‘(𝐼𝑘)) ∈ V) → (𝐵𝑘) = (2nd ‘(𝐼𝑘)))
183, 15, 17syl2anc 582 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) = (2nd ‘(𝐼𝑘)))
1918eqcomd 2731 . . . 4 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) = (𝐵𝑘))
2012, 19oveq12d 7437 . . 3 ((𝜑𝑘𝑋) → ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))) = ((𝐴𝑘)[,)(𝐵𝑘)))
214, 20eqtrd 2765 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
2221ixpeq2dva 8931 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  cmpt 5232   × cxp 5676  ccom 5682  wf 6545  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  Xcixp 8916  cr 11139  [,)cico 13361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-1st 7994  df-2nd 7995  df-ixp 8917
This theorem is referenced by:  opnvonmbllem2  46159
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