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Theorem sibff 29725
 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 29591 . . . 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 fvex 6113 . . . . . . 7 (TopOpen‘𝑊) ∈ V
76a1i 11 . . . . . 6 (𝜑 → (TopOpen‘𝑊) ∈ V)
85, 7syl5eqel 2692 . . . . 5 (𝜑𝐽 ∈ V)
98sgsiga 29532 . . . 4 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
104, 9syl5eqel 2692 . . 3 (𝜑𝑆 ran sigAlgebra)
11 sitgval.b . . . 4 𝐵 = (Base‘𝑊)
12 sitgval.0 . . . 4 0 = (0g𝑊)
13 sitgval.x . . . 4 · = ( ·𝑠𝑊)
14 sitgval.h . . . 4 𝐻 = (ℝHom‘(Scalar‘𝑊))
15 sitgval.1 . . . 4 (𝜑𝑊𝑉)
16 sibfmbl.1 . . . 4 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
1711, 5, 4, 12, 13, 14, 15, 1, 16sibfmbl 29724 . . 3 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
183, 10, 17mbfmf 29644 . 2 (𝜑𝐹: dom 𝑀 𝑆)
194unieqi 4381 . . . 4 𝑆 = (sigaGen‘𝐽)
20 unisg 29533 . . . . 5 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
218, 20syl 17 . . . 4 (𝜑 (sigaGen‘𝐽) = 𝐽)
2219, 21syl5eq 2656 . . 3 (𝜑 𝑆 = 𝐽)
2322feq3d 5945 . 2 (𝜑 → (𝐹: dom 𝑀 𝑆𝐹: dom 𝑀 𝐽))
2418, 23mpbid 221 1 (𝜑𝐹: dom 𝑀 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∪ cuni 4372  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  TopOpenctopn 15905  0gc0g 15923  ℝHomcrrh 29365  sigAlgebracsiga 29497  sigaGencsigagen 29528  measurescmeas 29585  sitgcsitg 29718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-esum 29417  df-siga 29498  df-sigagen 29529  df-meas 29586  df-mbfm 29640  df-sitg 29719 This theorem is referenced by:  sibfinima  29728  sibfof  29729  sitgaddlemb  29737  sitmcl  29740
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