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Theorem hoi2toco 46133
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 6593 . . . . . 6 Fun 𝐼
43a1i 11 . . . . 5 (𝜑 → Fun 𝐼)
54adantr 479 . . . 4 ((𝜑𝑘𝑋) → Fun 𝐼)
6 simpr 483 . . . . 5 ((𝜑𝑘𝑋) → 𝑘𝑋)
72dmeqi 5907 . . . . . . . 8 dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
87a1i 11 . . . . . . 7 (𝜑 → dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
9 opex 5466 . . . . . . . . . 10 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V
1092a1i 12 . . . . . . . . 9 (𝜑 → (𝑘𝑋 → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V))
111, 10ralrimi 3244 . . . . . . . 8 (𝜑 → ∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
12 dmmptg 6248 . . . . . . . 8 (∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
1311, 12syl 17 . . . . . . 7 (𝜑 → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
148, 13eqtr2d 2766 . . . . . 6 (𝜑𝑋 = dom 𝐼)
1514adantr 479 . . . . 5 ((𝜑𝑘𝑋) → 𝑋 = dom 𝐼)
166, 15eleqtrd 2827 . . . 4 ((𝜑𝑘𝑋) → 𝑘 ∈ dom 𝐼)
17 fvco 6995 . . . 4 ((Fun 𝐼𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
185, 16, 17syl2anc 582 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
199a1i 11 . . . . 5 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
202fvmpt2 7015 . . . . 5 ((𝑘𝑋 ∧ ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
216, 19, 20syl2anc 582 . . . 4 ((𝜑𝑘𝑋) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
2221fveq2d 6900 . . 3 ((𝜑𝑘𝑋) → ([,)‘(𝐼𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
23 df-ov 7422 . . . . 5 ((𝐴𝑘)[,)(𝐵𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩)
2423eqcomi 2734 . . . 4 ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘))
2524a1i 11 . . 3 ((𝜑𝑘𝑋) → ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘)))
2618, 22, 253eqtrd 2769 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
271, 26ixpeq2d 44574 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wnf 1777  wcel 2098  wral 3050  Vcvv 3461  cop 4636  cmpt 5232  dom cdm 5678  ccom 5682  Fun wfun 6543  cfv 6549  (class class class)co 7419  Xcixp 8916  [,)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
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-fv 6557  df-ov 7422  df-ixp 8917
This theorem is referenced by:  opnvonmbllem1  46158
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