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Theorem hoidifhspval2 45009
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 = (π‘₯ ∈ ℝ ↦ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘₯ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘₯), (π‘Žβ€˜π‘˜)))))
hoidifhspval2.y (πœ‘ β†’ π‘Œ ∈ ℝ)
hoidifhspval2.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
hoidifhspval2.a (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)
Assertion
Ref Expression
hoidifhspval2 (πœ‘ β†’ ((π·β€˜π‘Œ)β€˜π΄) = (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))))
Distinct variable groups:   𝐴,π‘Ž,π‘˜   𝐾,π‘Ž,π‘₯   𝑋,π‘Ž,π‘˜,π‘₯   π‘Œ,π‘Ž,π‘˜,π‘₯   πœ‘,π‘Ž,π‘₯
Allowed substitution hints:   πœ‘(π‘˜)   𝐴(π‘₯)   𝐷(π‘₯,π‘˜,π‘Ž)   𝐾(π‘˜)   𝑉(π‘₯,π‘˜,π‘Ž)

Proof of Theorem hoidifhspval2
StepHypRef Expression
1 hoidifhspval2.d . . 3 𝐷 = (π‘₯ ∈ ℝ ↦ (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘₯ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘₯), (π‘Žβ€˜π‘˜)))))
2 hoidifhspval2.y . . 3 (πœ‘ β†’ π‘Œ ∈ ℝ)
31, 2hoidifhspval 45002 . 2 (πœ‘ β†’ (π·β€˜π‘Œ) = (π‘Ž ∈ (ℝ ↑m 𝑋) ↦ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘Œ), (π‘Žβ€˜π‘˜)))))
4 fveq1 6861 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘Žβ€˜π‘˜) = (π΄β€˜π‘˜))
54breq2d 5137 . . . . . 6 (π‘Ž = 𝐴 β†’ (π‘Œ ≀ (π‘Žβ€˜π‘˜) ↔ π‘Œ ≀ (π΄β€˜π‘˜)))
65, 4ifbieq1d 4530 . . . . 5 (π‘Ž = 𝐴 β†’ if(π‘Œ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘Œ) = if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ))
76, 4ifeq12d 4527 . . . 4 (π‘Ž = 𝐴 β†’ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘Œ), (π‘Žβ€˜π‘˜)) = if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜)))
87mpteq2dv 5227 . . 3 (π‘Ž = 𝐴 β†’ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘Œ), (π‘Žβ€˜π‘˜))) = (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))))
98adantl 482 . 2 ((πœ‘ ∧ π‘Ž = 𝐴) β†’ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π‘Žβ€˜π‘˜), (π‘Žβ€˜π‘˜), π‘Œ), (π‘Žβ€˜π‘˜))) = (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))))
10 hoidifhspval2.a . . 3 (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)
11 reex 11166 . . . . . 6 ℝ ∈ V
1211a1i 11 . . . . 5 (πœ‘ β†’ ℝ ∈ V)
13 hoidifhspval2.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑉)
1412, 13jca 512 . . . 4 (πœ‘ β†’ (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
15 elmapg 8800 . . . 4 ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) β†’ (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:π‘‹βŸΆβ„))
1614, 15syl 17 . . 3 (πœ‘ β†’ (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:π‘‹βŸΆβ„))
1710, 16mpbird 256 . 2 (πœ‘ β†’ 𝐴 ∈ (ℝ ↑m 𝑋))
18 mptexg 7191 . . 3 (𝑋 ∈ 𝑉 β†’ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))) ∈ V)
1913, 18syl 17 . 2 (πœ‘ β†’ (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))) ∈ V)
203, 9, 17, 19fvmptd 6975 1 (πœ‘ β†’ ((π·β€˜π‘Œ)β€˜π΄) = (π‘˜ ∈ 𝑋 ↦ if(π‘˜ = 𝐾, if(π‘Œ ≀ (π΄β€˜π‘˜), (π΄β€˜π‘˜), π‘Œ), (π΄β€˜π‘˜))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3459  ifcif 4506   class class class wbr 5125   ↦ cmpt 5208  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377   ↑m cmap 8787  β„cr 11074   ≀ cle 11214
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-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132
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-reu 3365  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-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  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-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-map 8789
This theorem is referenced by:  hoidifhspf  45012  hoidifhspval3  45013
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