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Theorem orvcval4 31832
 Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 31829. (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 31515 . . . . 5 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
4 orvccel.3 . . . . 5 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
51, 3, 4isanmbfm 31628 . . . 4 (𝜑𝑋 ran MblFnM)
65mbfmfun 31626 . . 3 (𝜑 → Fun 𝑋)
71, 3, 4mbfmf 31627 . . . . 5 (𝜑𝑋: 𝑆 (sigaGen‘𝐽))
8 elex 3462 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ V)
9 unisg 31516 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
102, 8, 93syl 18 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
1110feq3d 6478 . . . . 5 (𝜑 → (𝑋: 𝑆 (sigaGen‘𝐽) ↔ 𝑋: 𝑆 𝐽))
127, 11mpbid 235 . . . 4 (𝜑𝑋: 𝑆 𝐽)
1312frnd 6498 . . 3 (𝜑 → ran 𝑋 𝐽)
14 fimacnvinrn2 6822 . . 3 ((Fun 𝑋 ∧ ran 𝑋 𝐽) → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
156, 13, 14syl2anc 587 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
16 orvccel.4 . . 3 (𝜑𝐴𝑉)
176, 4, 16orvcval 31829 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
18 dfrab2 4234 . . . 4 {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽)
1918a1i 11 . . 3 (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽))
2019imaeq2d 5900 . 2 (𝜑 → (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
2115, 17, 203eqtr4d 2846 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  {cab 2779  {crab 3113  Vcvv 3444   ∩ cin 3883   ⊆ wss 3884  ∪ cuni 4803   class class class wbr 5033  ◡ccnv 5522  ran crn 5524   “ cima 5526  Fun wfun 6322  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  Topctop 21502  sigAlgebracsiga 31481  sigaGencsigagen 31511  MblFnMcmbfm 31622  ∘RV/𝑐corvc 31827 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-siga 31482  df-sigagen 31512  df-mbfm 31623  df-orvc 31828 This theorem is referenced by:  orvcoel  31833  orvccel  31834  orrvcval4  31836
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