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Theorem orvcval4 34464
Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 34461. (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.3 . . . . 5 (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
21isanmbfm 34259 . . . 4 (𝜑𝑋 ran MblFnM)
32mbfmfun 34255 . . 3 (𝜑 → Fun 𝑋)
4 orvccel.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
5 orvccel.2 . . . . . . 7 (𝜑𝐽 ∈ Top)
65sgsiga 34144 . . . . . 6 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
74, 6, 1mbfmf 34256 . . . . 5 (𝜑𝑋: 𝑆 (sigaGen‘𝐽))
8 elex 3500 . . . . . . 7 (𝐽 ∈ Top → 𝐽 ∈ V)
9 unisg 34145 . . . . . . 7 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
105, 8, 93syl 18 . . . . . 6 (𝜑 (sigaGen‘𝐽) = 𝐽)
1110feq3d 6722 . . . . 5 (𝜑 → (𝑋: 𝑆 (sigaGen‘𝐽) ↔ 𝑋: 𝑆 𝐽))
127, 11mpbid 232 . . . 4 (𝜑𝑋: 𝑆 𝐽)
1312frnd 6743 . . 3 (𝜑 → ran 𝑋 𝐽)
14 fimacnvinrn2 7091 . . 3 ((Fun 𝑋 ∧ ran 𝑋 𝐽) → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
153, 13, 14syl2anc 584 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
16 orvccel.4 . . 3 (𝜑𝐴𝑉)
173, 1, 16orvcval 34461 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
18 dfrab2 4319 . . . 4 {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽)
1918a1i 11 . . 3 (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} = ({𝑦𝑦𝑅𝐴} ∩ 𝐽))
2019imaeq2d 6077 . 2 (𝜑 → (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}) = (𝑋 “ ({𝑦𝑦𝑅𝐴} ∩ 𝐽)))
2115, 17, 203eqtr4d 2786 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {cab 2713  {crab 3435  Vcvv 3479  cin 3949  wss 3950   cuni 4906   class class class wbr 5142  ccnv 5683  ran crn 5685  cima 5687  Fun wfun 6554  wf 6556  cfv 6560  (class class class)co 7432  Topctop 22900  sigAlgebracsiga 34110  sigaGencsigagen 34140  MblFnMcmbfm 34251  RV/𝑐corvc 34459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-siga 34111  df-sigagen 34141  df-mbfm 34252  df-orvc 34460
This theorem is referenced by:  orvcoel  34465  orvccel  34466  orrvcval4  34468
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