Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sibff Structured version   Visualization version   GIF version

Theorem sibff 29531
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 29397 . . . 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 6098 . . . . . . 7 (TopOpen‘𝑊) ∈ V
76a1i 11 . . . . . 6 (𝜑 → (TopOpen‘𝑊) ∈ V)
85, 7syl5eqel 2691 . . . . 5 (𝜑𝐽 ∈ V)
98sgsiga 29338 . . . 4 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
104, 9syl5eqel 2691 . . 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 29530 . . 3 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
183, 10, 17mbfmf 29450 . 2 (𝜑𝐹: dom 𝑀 𝑆)
194unieqi 4375 . . . 4 𝑆 = (sigaGen‘𝐽)
20 unisg 29339 . . . . 5 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
218, 20syl 17 . . . 4 (𝜑 (sigaGen‘𝐽) = 𝐽)
2219, 21syl5eq 2655 . . 3 (𝜑 𝑆 = 𝐽)
2322feq3d 5931 . 2 (𝜑 → (𝐹: dom 𝑀 𝑆𝐹: dom 𝑀 𝐽))
2418, 23mpbid 220 1 (𝜑𝐹: dom 𝑀 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172   cuni 4366  dom cdm 5028  ran crn 5029  wf 5786  cfv 5790  (class class class)co 6527  Basecbs 15641  Scalarcsca 15717   ·𝑠 cvsca 15718  TopOpenctopn 15851  0gc0g 15869  ℝHomcrrh 29171  sigAlgebracsiga 29303  sigaGencsigagen 29334  measurescmeas 29391  sitgcsitg 29524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-esum 29223  df-siga 29304  df-sigagen 29335  df-meas 29392  df-mbfm 29446  df-sitg 29525
This theorem is referenced by:  sibfinima  29534  sibfof  29535  sitgaddlemb  29543  sitmcl  29546
  Copyright terms: Public domain W3C validator