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Theorem orvcval4 34442
Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 34439. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Hypotheses
Ref Expression
orvccel.1 (𝜑𝑆 ran sigAlgebra)
orvccel.2 (𝜑𝐽 ∈ Top)
orvccel.3 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
orvccel.4 (𝜑𝐴𝑉)
Assertion
Ref Expression
orvcval4 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋   𝑦,𝐽
Allowed substitution hints:   𝜑(𝑦)   𝑆(𝑦)   𝑉(𝑦)

Proof of Theorem orvcval4
StepHypRef Expression
1 orvccel.3 . . . . 5 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
21isanmbfm 34238 . . . 4 (𝜑𝑋 ran MblFnM)
32mbfmfun 34234 . . 3 (𝜑 → Fun 𝑋)
4 orvccel.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
5 orvccel.2 . . . . . . 7 (𝜑𝐽 ∈ Top)
65sgsiga 34123 . . . . . 6 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
74, 6, 1mbfmf 34235 . . . . 5 (𝜑𝑋: 𝑆 (sigaGen‘𝐽))
8 elex 3499 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ V)
9 unisg 34124 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
105, 8, 93syl 18 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
1110feq3d 6724 . . . . 5 (𝜑 → (𝑋: 𝑆 (sigaGen‘𝐽) ↔ 𝑋: 𝑆 𝐽))
127, 11mpbid 232 . . . 4 (𝜑𝑋: 𝑆 𝐽)
1312frnd 6745 . . 3 (𝜑 → ran 𝑋 𝐽)
14 fimacnvinrn2 7092 . . 3 ((Fun 𝑋 ∧ ran 𝑋 𝐽) → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
153, 13, 14syl2anc 584 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
16 orvccel.4 . . 3 (𝜑𝐴𝑉)
173, 1, 16orvcval 34439 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
18 dfrab2 4326 . . . 4 {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽)
1918a1i 11 . . 3 (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽))
2019imaeq2d 6080 . 2 (𝜑 → (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
2115, 17, 203eqtr4d 2785 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478  cin 3962  wss 3963   cuni 4912   class class class wbr 5148  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  sigAlgebracsiga 34089  sigaGencsigagen 34119  MblFnMcmbfm 34230  RV/𝑐corvc 34437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-siga 34090  df-sigagen 34120  df-mbfm 34231  df-orvc 34438
This theorem is referenced by:  orvcoel  34443  orvccel  34444  orrvcval4  34446
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