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Theorem hoi2toco 45001
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 6560 . . . . . 6 Fun 𝐼
43a1i 11 . . . . 5 (𝜑 → Fun 𝐼)
54adantr 481 . . . 4 ((𝜑𝑘𝑋) → Fun 𝐼)
6 simpr 485 . . . . 5 ((𝜑𝑘𝑋) → 𝑘𝑋)
72dmeqi 5880 . . . . . . . 8 dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
87a1i 11 . . . . . . 7 (𝜑 → dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
9 opex 5441 . . . . . . . . . 10 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V
1092a1i 12 . . . . . . . . 9 (𝜑 → (𝑘𝑋 → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V))
111, 10ralrimi 3251 . . . . . . . 8 (𝜑 → ∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
12 dmmptg 6214 . . . . . . . 8 (∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
1311, 12syl 17 . . . . . . 7 (𝜑 → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
148, 13eqtr2d 2772 . . . . . 6 (𝜑𝑋 = dom 𝐼)
1514adantr 481 . . . . 5 ((𝜑𝑘𝑋) → 𝑋 = dom 𝐼)
166, 15eleqtrd 2834 . . . 4 ((𝜑𝑘𝑋) → 𝑘 ∈ dom 𝐼)
17 fvco 6959 . . . 4 ((Fun 𝐼𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
185, 16, 17syl2anc 584 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
199a1i 11 . . . . 5 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
202fvmpt2 6979 . . . . 5 ((𝑘𝑋 ∧ ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
216, 19, 20syl2anc 584 . . . 4 ((𝜑𝑘𝑋) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
2221fveq2d 6866 . . 3 ((𝜑𝑘𝑋) → ([,)‘(𝐼𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
23 df-ov 7380 . . . . 5 ((𝐴𝑘)[,)(𝐵𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩)
2423eqcomi 2740 . . . 4 ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘))
2524a1i 11 . . 3 ((𝜑𝑘𝑋) → ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘)))
2618, 22, 253eqtrd 2775 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
271, 26ixpeq2d 43431 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  wral 3060  Vcvv 3459  cop 4612  cmpt 5208  dom cdm 5653  ccom 5657  Fun wfun 6510  cfv 6516  (class class class)co 7377  Xcixp 8857  [,)cico 13291
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-fv 6524  df-ov 7380  df-ixp 8858
This theorem is referenced by:  opnvonmbllem1  45026
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