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Theorem hoi2toco 39497
 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 𝑘𝜑
hoi2toco.c 𝐼 = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
Assertion
Ref Expression
hoi2toco (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Distinct variable group:   𝑘,𝑋
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐵(𝑘)   𝐼(𝑘)

Proof of Theorem hoi2toco
StepHypRef Expression
1 hoi2toco.1 . 2 𝑘𝜑
2 hoi2toco.c . . . . . . 7 𝐼 = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
32funmpt2 5841 . . . . . 6 Fun 𝐼
43a1i 11 . . . . 5 (𝜑 → Fun 𝐼)
54adantr 480 . . . 4 ((𝜑𝑘𝑋) → Fun 𝐼)
6 simpr 476 . . . . 5 ((𝜑𝑘𝑋) → 𝑘𝑋)
72dmeqi 5247 . . . . . . . 8 dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
87a1i 11 . . . . . . 7 (𝜑 → dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
9 opex 4859 . . . . . . . . . 10 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V
1092a1i 12 . . . . . . . . 9 (𝜑 → (𝑘𝑋 → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V))
111, 10ralrimi 2940 . . . . . . . 8 (𝜑 → ∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
12 dmmptg 5549 . . . . . . . 8 (∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
1311, 12syl 17 . . . . . . 7 (𝜑 → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
148, 13eqtr2d 2645 . . . . . 6 (𝜑𝑋 = dom 𝐼)
1514adantr 480 . . . . 5 ((𝜑𝑘𝑋) → 𝑋 = dom 𝐼)
166, 15eleqtrd 2690 . . . 4 ((𝜑𝑘𝑋) → 𝑘 ∈ dom 𝐼)
17 fvco 6184 . . . 4 ((Fun 𝐼𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
185, 16, 17syl2anc 691 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
199a1i 11 . . . . 5 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
202fvmpt2 6200 . . . . 5 ((𝑘𝑋 ∧ ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
216, 19, 20syl2anc 691 . . . 4 ((𝜑𝑘𝑋) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
2221fveq2d 6107 . . 3 ((𝜑𝑘𝑋) → ([,)‘(𝐼𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
23 df-ov 6552 . . . . 5 ((𝐴𝑘)[,)(𝐵𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩)
2423eqcomi 2619 . . . 4 ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘))
2524a1i 11 . . 3 ((𝜑𝑘𝑋) → ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘)))
2618, 22, 253eqtrd 2648 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
271, 26ixpeq2d 38262 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  dom cdm 5038   ∘ ccom 5042  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  Xcixp 7794  [,)cico 12048 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552  df-ixp 7795 This theorem is referenced by:  opnvonmbllem1  39522
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