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Theorem orvcval4 31064
Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 31061. (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.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
2 orvccel.2 . . . . . 6 (𝜑𝐽 ∈ Top)
32sgsiga 30746 . . . . 5 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
4 orvccel.3 . . . . 5 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
51, 3, 4isanmbfm 30859 . . . 4 (𝜑𝑋 ran MblFnM)
65mbfmfun 30857 . . 3 (𝜑 → Fun 𝑋)
71, 3, 4mbfmf 30858 . . . . 5 (𝜑𝑋: 𝑆 (sigaGen‘𝐽))
8 elex 3429 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ V)
9 unisg 30747 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
102, 8, 93syl 18 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
1110feq3d 6269 . . . . 5 (𝜑 → (𝑋: 𝑆 (sigaGen‘𝐽) ↔ 𝑋: 𝑆 𝐽))
127, 11mpbid 224 . . . 4 (𝜑𝑋: 𝑆 𝐽)
1312frnd 6289 . . 3 (𝜑 → ran 𝑋 𝐽)
14 fimacnvinrn2 6603 . . 3 ((Fun 𝑋 ∧ ran 𝑋 𝐽) → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
156, 13, 14syl2anc 579 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
16 orvccel.4 . . 3 (𝜑𝐴𝑉)
176, 4, 16orvcval 31061 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
18 dfrab2 4134 . . . 4 {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽)
1918a1i 11 . . 3 (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽))
2019imaeq2d 5711 . 2 (𝜑 → (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
2115, 17, 203eqtr4d 2871 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  {cab 2811  {crab 3121  Vcvv 3414  cin 3797  wss 3798   cuni 4660   class class class wbr 4875  ccnv 5345  ran crn 5347  cima 5349  Fun wfun 6121  wf 6123  cfv 6127  (class class class)co 6910  Topctop 21075  sigAlgebracsiga 30711  sigaGencsigagen 30742  MblFnMcmbfm 30853  RV/𝑐corvc 31059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fo 6133  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-siga 30712  df-sigagen 30743  df-mbfm 30854  df-orvc 31060
This theorem is referenced by:  orvcoel  31065  orvccel  31066  orrvcval4  31068
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