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Theorem hoi2toco 39301
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 5827 . . . . . 6 Fun 𝐼
43a1i 11 . . . . 5 (𝜑 → Fun 𝐼)
54adantr 479 . . . 4 ((𝜑𝑘𝑋) → Fun 𝐼)
6 simpr 475 . . . . 5 ((𝜑𝑘𝑋) → 𝑘𝑋)
72dmeqi 5234 . . . . . . . 8 dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
87a1i 11 . . . . . . 7 (𝜑 → dom 𝐼 = dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
9 opex 4853 . . . . . . . . . 10 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V
1092a1i 12 . . . . . . . . 9 (𝜑 → (𝑘𝑋 → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V))
111, 10ralrimi 2939 . . . . . . . 8 (𝜑 → ∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
12 dmmptg 5535 . . . . . . . 8 (∀𝑘𝑋 ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
1311, 12syl 17 . . . . . . 7 (𝜑 → dom (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = 𝑋)
148, 13eqtr2d 2644 . . . . . 6 (𝜑𝑋 = dom 𝐼)
1514adantr 479 . . . . 5 ((𝜑𝑘𝑋) → 𝑋 = dom 𝐼)
166, 15eleqtrd 2689 . . . 4 ((𝜑𝑘𝑋) → 𝑘 ∈ dom 𝐼)
17 fvco 6169 . . . 4 ((Fun 𝐼𝑘 ∈ dom 𝐼) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
185, 16, 17syl2anc 690 . . 3 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ([,)‘(𝐼𝑘)))
199a1i 11 . . . . 5 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
202fvmpt2 6185 . . . . 5 ((𝑘𝑋 ∧ ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
216, 19, 20syl2anc 690 . . . 4 ((𝜑𝑘𝑋) → (𝐼𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
2221fveq2d 6092 . . 3 ((𝜑𝑘𝑋) → ([,)‘(𝐼𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
23 df-ov 6530 . . . . 5 ((𝐴𝑘)[,)(𝐵𝑘)) = ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩)
2423eqcomi 2618 . . . 4 ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘))
2524a1i 11 . . 3 ((𝜑𝑘𝑋) → ([,)‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = ((𝐴𝑘)[,)(𝐵𝑘)))
2618, 22, 253eqtrd 2647 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
271, 26ixpeq2d 38065 1 (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wnf 1698  wcel 1976  wral 2895  Vcvv 3172  cop 4130  cmpt 4637  dom cdm 5028  ccom 5032  Fun wfun 5784  cfv 5790  (class class class)co 6527  Xcixp 7771  [,)cico 12004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798  df-ov 6530  df-ixp 7772
This theorem is referenced by:  opnvonmbllem1  39326
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