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Theorem orvcval4 29849
 Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 29846. (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 29532 . . . . 5 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
4 orvccel.3 . . . . 5 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
51, 3, 4isanmbfm 29645 . . . 4 (𝜑𝑋 ran MblFnM)
65mbfmfun 29643 . . 3 (𝜑 → Fun 𝑋)
71, 3, 4mbfmf 29644 . . . . 5 (𝜑𝑋: 𝑆 (sigaGen‘𝐽))
8 elex 3185 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ V)
9 unisg 29533 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
102, 8, 93syl 18 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
1110feq3d 5945 . . . . 5 (𝜑 → (𝑋: 𝑆 (sigaGen‘𝐽) ↔ 𝑋: 𝑆 𝐽))
127, 11mpbid 221 . . . 4 (𝜑𝑋: 𝑆 𝐽)
13 frn 5966 . . . 4 (𝑋: 𝑆 𝐽 → ran 𝑋 𝐽)
1412, 13syl 17 . . 3 (𝜑 → ran 𝑋 𝐽)
15 fimacnvinrn2 6257 . . 3 ((Fun 𝑋 ∧ ran 𝑋 𝐽) → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
166, 14, 15syl2anc 691 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
17 orvccel.4 . . 3 (𝜑𝐴𝑉)
186, 4, 17orvcval 29846 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
19 dfrab2 3862 . . . 4 {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽)
2019a1i 11 . . 3 (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽))
2120imaeq2d 5385 . 2 (𝜑 → (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
2216, 18, 213eqtr4d 2654 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  ◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Topctop 20517  sigAlgebracsiga 29497  sigaGencsigagen 29528  MblFnMcmbfm 29639  ∘RV/𝑐corvc 29844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-siga 29498  df-sigagen 29529  df-mbfm 29640  df-orvc 29845 This theorem is referenced by:  orvcoel  29850  orvccel  29851  orrvcval4  29853
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