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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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