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Theorem hoidifhspval2 40166
 Description: 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
hoidifhspval2.d 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
hoidifhspval2.y (𝜑𝑌 ∈ ℝ)
hoidifhspval2.x (𝜑𝑋𝑉)
hoidifhspval2.a (𝜑𝐴:𝑋⟶ℝ)
Assertion
Ref Expression
hoidifhspval2 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
Distinct variable groups:   𝐴,𝑎,𝑘   𝐾,𝑎,𝑥   𝑋,𝑎,𝑘,𝑥   𝑌,𝑎,𝑘,𝑥   𝜑,𝑎,𝑥
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑥)   𝐷(𝑥,𝑘,𝑎)   𝐾(𝑘)   𝑉(𝑥,𝑘,𝑎)

Proof of Theorem hoidifhspval2
StepHypRef Expression
1 hoidifhspval2.d . . 3 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
2 hoidifhspval2.y . . 3 (𝜑𝑌 ∈ ℝ)
31, 2hoidifhspval 40159 . 2 (𝜑 → (𝐷𝑌) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎𝑘), (𝑎𝑘), 𝑌), (𝑎𝑘)))))
4 fveq1 6157 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑘) = (𝐴𝑘))
54breq2d 4635 . . . . . 6 (𝑎 = 𝐴 → (𝑌 ≤ (𝑎𝑘) ↔ 𝑌 ≤ (𝐴𝑘)))
65, 4ifbieq1d 4087 . . . . 5 (𝑎 = 𝐴 → if(𝑌 ≤ (𝑎𝑘), (𝑎𝑘), 𝑌) = if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌))
76, 4ifeq12d 4084 . . . 4 (𝑎 = 𝐴 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎𝑘), (𝑎𝑘), 𝑌), (𝑎𝑘)) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)))
87mpteq2dv 4715 . . 3 (𝑎 = 𝐴 → (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎𝑘), (𝑎𝑘), 𝑌), (𝑎𝑘))) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
98adantl 482 . 2 ((𝜑𝑎 = 𝐴) → (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎𝑘), (𝑎𝑘), 𝑌), (𝑎𝑘))) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
10 hoidifhspval2.a . . 3 (𝜑𝐴:𝑋⟶ℝ)
11 reex 9987 . . . . . 6 ℝ ∈ V
1211a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
13 hoidifhspval2.x . . . . 5 (𝜑𝑋𝑉)
1412, 13jca 554 . . . 4 (𝜑 → (ℝ ∈ V ∧ 𝑋𝑉))
15 elmapg 7830 . . . 4 ((ℝ ∈ V ∧ 𝑋𝑉) → (𝐴 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐴:𝑋⟶ℝ))
1614, 15syl 17 . . 3 (𝜑 → (𝐴 ∈ (ℝ ↑𝑚 𝑋) ↔ 𝐴:𝑋⟶ℝ))
1710, 16mpbird 247 . 2 (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))
18 mptexg 6449 . . 3 (𝑋𝑉 → (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))) ∈ V)
1913, 18syl 17 . 2 (𝜑 → (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))) ∈ V)
203, 9, 17, 19fvmptd 6255 1 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3190  ifcif 4064   class class class wbr 4623   ↦ cmpt 4683  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ↑𝑚 cmap 7817  ℝcr 9895   ≤ cle 10035 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819 This theorem is referenced by:  hoidifhspf  40169  hoidifhspval3  40170
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