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Theorem hoi2toco 43088
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 6375 . . . . . 6 Fun 𝐼
43a1i 11 . . . . 5 (𝜑 → Fun 𝐼)
54adantr 484 . . . 4 ((𝜑𝑘𝑋) → Fun 𝐼)
6 simpr 488 . . . . 5 ((𝜑𝑘𝑋) → 𝑘𝑋)
72dmeqi 5754 . . . . . . . 8 dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
87a1i 11 . . . . . . 7 (𝜑 → dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
9 opex 5337 . . . . . . . . . 10 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V
1092a1i 12 . . . . . . . . 9 (𝜑 → (𝑘𝑋 → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V))
111, 10ralrimi 3210 . . . . . . . 8 (𝜑 → ∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
12 dmmptg 6077 . . . . . . . 8 (∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
1311, 12syl 17 . . . . . . 7 (𝜑 → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
148, 13eqtr2d 2860 . . . . . 6 (𝜑𝑋 = dom 𝐼)
1514adantr 484 . . . . 5 ((𝜑𝑘𝑋) → 𝑋 = dom 𝐼)
166, 15eleqtrd 2918 . . . 4 ((𝜑𝑘𝑋) → 𝑘 ∈ dom 𝐼)
17 fvco 6740 . . . 4 ((Fun 𝐼𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
185, 16, 17syl2anc 587 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
199a1i 11 . . . . 5 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
202fvmpt2 6760 . . . . 5 ((𝑘𝑋 ∧ ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
216, 19, 20syl2anc 587 . . . 4 ((𝜑𝑘𝑋) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
2221fveq2d 6655 . . 3 ((𝜑𝑘𝑋) → ([,)‘(𝐼𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
23 df-ov 7141 . . . . 5 ((𝐴𝑘)[,)(𝐵𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩)
2423eqcomi 2833 . . . 4 ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘))
2524a1i 11 . . 3 ((𝜑𝑘𝑋) → ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘)))
2618, 22, 253eqtrd 2863 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
271, 26ixpeq2d 41538 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2115  wral 3132  Vcvv 3479  cop 4554  cmpt 5127  dom cdm 5536  ccom 5540  Fun wfun 6330  cfv 6336  (class class class)co 7138  Xcixp 8444  [,)cico 12726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-ov 7141  df-ixp 8445
This theorem is referenced by:  opnvonmbllem1  43113
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