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Theorem sibfima 29529
Description: Any preimage of a singleton by a simple function is measurable. (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
sibfima ((𝜑𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝐴})) ∈ (0[,)+∞))

Proof of Theorem sibfima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sibfmbl.1 . . . 4 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
2 sitgval.b . . . . 5 𝐵 = (Base‘𝑊)
3 sitgval.j . . . . 5 𝐽 = (TopOpen‘𝑊)
4 sitgval.s . . . . 5 𝑆 = (sigaGen‘𝐽)
5 sitgval.0 . . . . 5 0 = (0g𝑊)
6 sitgval.x . . . . 5 · = ( ·𝑠𝑊)
7 sitgval.h . . . . 5 𝐻 = (ℝHom‘(Scalar‘𝑊))
8 sitgval.1 . . . . 5 (𝜑𝑊𝑉)
9 sitgval.2 . . . . 5 (𝜑𝑀 ran measures)
102, 3, 4, 5, 6, 7, 8, 9issibf 29524 . . . 4 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
111, 10mpbid 220 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
1211simp3d 1067 . 2 (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))
13 sneq 4130 . . . . . 6 (𝑥 = 𝐴 → {𝑥} = {𝐴})
1413imaeq2d 5368 . . . . 5 (𝑥 = 𝐴 → (𝐹 “ {𝑥}) = (𝐹 “ {𝐴}))
1514fveq2d 6088 . . . 4 (𝑥 = 𝐴 → (𝑀‘(𝐹 “ {𝑥})) = (𝑀‘(𝐹 “ {𝐴})))
1615eleq1d 2667 . . 3 (𝑥 = 𝐴 → ((𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝐹 “ {𝐴})) ∈ (0[,)+∞)))
1716rspcv 3273 . 2 (𝐴 ∈ (ran 𝐹 ∖ { 0 }) → (∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞) → (𝑀‘(𝐹 “ {𝐴})) ∈ (0[,)+∞)))
1812, 17mpan9 484 1 ((𝜑𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝐴})) ∈ (0[,)+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2891  cdif 3532  {csn 4120   cuni 4362  ccnv 5023  dom cdm 5024  ran crn 5025  cima 5027  cfv 5786  (class class class)co 6523  Fincfn 7814  0cc0 9788  +∞cpnf 9923  [,)cico 12000  Basecbs 15637  Scalarcsca 15713   ·𝑠 cvsca 15714  TopOpenctopn 15847  0gc0g 15865  ℝHomcrrh 29167  sigaGencsigagen 29330  measurescmeas 29387  MblFnMcmbfm 29441  sitgcsitg 29520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-sitg 29521
This theorem is referenced by:  sibfinima  29530  sitgfval  29532  sitgclg  29533
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