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Theorem sibff 34318
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 34182 . . . 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 6922 . . . . . 6 (𝜑 → (TopOpen‘𝑊) ∈ V)
75, 6eqeltrid 2843 . . . . 5 (𝜑𝐽 ∈ V)
87sgsiga 34123 . . . 4 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
94, 8eqeltrid 2843 . . 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 34317 . . 3 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
173, 9, 16mbfmf 34235 . 2 (𝜑𝐹: dom 𝑀 𝑆)
184unieqi 4924 . . . 4 𝑆 = (sigaGen‘𝐽)
19 unisg 34124 . . . . 5 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
207, 19syl 17 . . . 4 (𝜑 (sigaGen‘𝐽) = 𝐽)
2118, 20eqtrid 2787 . . 3 (𝜑 𝑆 = 𝐽)
2221feq3d 6724 . 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 1537  wcel 2106  Vcvv 3478   cuni 4912  dom cdm 5689  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  TopOpenctopn 17468  0gc0g 17486  ℝHomcrrh 33956  sigAlgebracsiga 34089  sigaGencsigagen 34119  measurescmeas 34176  sitgcsitg 34311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-esum 34009  df-siga 34090  df-sigagen 34120  df-meas 34177  df-mbfm 34231  df-sitg 34312
This theorem is referenced by:  sibfinima  34321  sibfof  34322  sitgaddlemb  34330  sitmcl  34333
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