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Theorem orvcval4 30856
Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 30853. (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 30539 . . . . 5 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
4 orvccel.3 . . . . 5 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
51, 3, 4isanmbfm 30652 . . . 4 (𝜑𝑋 ran MblFnM)
65mbfmfun 30650 . . 3 (𝜑 → Fun 𝑋)
71, 3, 4mbfmf 30651 . . . . 5 (𝜑𝑋: 𝑆 (sigaGen‘𝐽))
8 elex 3361 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ V)
9 unisg 30540 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
102, 8, 93syl 18 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
1110feq3d 6172 . . . . 5 (𝜑 → (𝑋: 𝑆 (sigaGen‘𝐽) ↔ 𝑋: 𝑆 𝐽))
127, 11mpbid 222 . . . 4 (𝜑𝑋: 𝑆 𝐽)
13 frn 6193 . . . 4 (𝑋: 𝑆 𝐽 → ran 𝑋 𝐽)
1412, 13syl 17 . . 3 (𝜑 → ran 𝑋 𝐽)
15 fimacnvinrn2 6492 . . 3 ((Fun 𝑋 ∧ ran 𝑋 𝐽) → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
166, 14, 15syl2anc 565 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
17 orvccel.4 . . 3 (𝜑𝐴𝑉)
186, 4, 17orvcval 30853 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
19 dfrab2 4049 . . . 4 {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽)
2019a1i 11 . . 3 (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽))
2120imaeq2d 5607 . 2 (𝜑 → (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
2216, 18, 213eqtr4d 2814 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  {cab 2756  {crab 3064  Vcvv 3349  cin 3720  wss 3721   cuni 4572   class class class wbr 4784  ccnv 5248  ran crn 5250  cima 5252  Fun wfun 6025  wf 6027  cfv 6031  (class class class)co 6792  Topctop 20917  sigAlgebracsiga 30504  sigaGencsigagen 30535  MblFnMcmbfm 30646  RV/𝑐corvc 30851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-siga 30505  df-sigagen 30536  df-mbfm 30647  df-orvc 30852
This theorem is referenced by:  orvcoel  30857  orvccel  30858  orrvcval4  30860
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