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Theorem sibff 34338
Description: A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
Hypotheses
Ref Expression
sitgval.b 𝐵 = (Base‘𝑊)
sitgval.j 𝐽 = (TopOpen‘𝑊)
sitgval.s 𝑆 = (sigaGen‘𝐽)
sitgval.0 0 = (0g𝑊)
sitgval.x · = ( ·𝑠𝑊)
sitgval.h 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1 (𝜑𝑊𝑉)
sitgval.2 (𝜑𝑀 ran measures)
sibfmbl.1 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
Assertion
Ref Expression
sibff (𝜑𝐹: dom 𝑀 𝐽)

Proof of Theorem sibff
StepHypRef Expression
1 sitgval.2 . . . 4 (𝜑𝑀 ran measures)
2 dmmeas 34202 . . . 4 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
31, 2syl 17 . . 3 (𝜑 → dom 𝑀 ran sigAlgebra)
4 sitgval.s . . . 4 𝑆 = (sigaGen‘𝐽)
5 sitgval.j . . . . . 6 𝐽 = (TopOpen‘𝑊)
6 fvexd 6921 . . . . . 6 (𝜑 → (TopOpen‘𝑊) ∈ V)
75, 6eqeltrid 2845 . . . . 5 (𝜑𝐽 ∈ V)
87sgsiga 34143 . . . 4 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
94, 8eqeltrid 2845 . . 3 (𝜑𝑆 ran sigAlgebra)
10 sitgval.b . . . 4 𝐵 = (Base‘𝑊)
11 sitgval.0 . . . 4 0 = (0g𝑊)
12 sitgval.x . . . 4 · = ( ·𝑠𝑊)
13 sitgval.h . . . 4 𝐻 = (ℝHom‘(Scalar‘𝑊))
14 sitgval.1 . . . 4 (𝜑𝑊𝑉)
15 sibfmbl.1 . . . 4 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
1610, 5, 4, 11, 12, 13, 14, 1, 15sibfmbl 34337 . . 3 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
173, 9, 16mbfmf 34255 . 2 (𝜑𝐹: dom 𝑀 𝑆)
184unieqi 4919 . . . 4 𝑆 = (sigaGen‘𝐽)
19 unisg 34144 . . . . 5 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
207, 19syl 17 . . . 4 (𝜑 (sigaGen‘𝐽) = 𝐽)
2118, 20eqtrid 2789 . . 3 (𝜑 𝑆 = 𝐽)
2221feq3d 6723 . 2 (𝜑 → (𝐹: dom 𝑀 𝑆𝐹: dom 𝑀 𝐽))
2317, 22mpbid 232 1 (𝜑𝐹: dom 𝑀 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480   cuni 4907  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  TopOpenctopn 17466  0gc0g 17484  ℝHomcrrh 33994  sigAlgebracsiga 34109  sigaGencsigagen 34139  measurescmeas 34196  sitgcsitg 34331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-esum 34029  df-siga 34110  df-sigagen 34140  df-meas 34197  df-mbfm 34251  df-sitg 34332
This theorem is referenced by:  sibfinima  34341  sibfof  34342  sitgaddlemb  34350  sitmcl  34353
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