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Theorem sibff 32544
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 32408 . . . 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 6834 . . . . . 6 (𝜑 → (TopOpen‘𝑊) ∈ V)
75, 6eqeltrid 2841 . . . . 5 (𝜑𝐽 ∈ V)
87sgsiga 32349 . . . 4 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
94, 8eqeltrid 2841 . . 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 32543 . . 3 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
173, 9, 16mbfmf 32461 . 2 (𝜑𝐹: dom 𝑀 𝑆)
184unieqi 4864 . . . 4 𝑆 = (sigaGen‘𝐽)
19 unisg 32350 . . . . 5 (𝐽 ∈ V → (sigaGen‘𝐽) = 𝐽)
207, 19syl 17 . . . 4 (𝜑 (sigaGen‘𝐽) = 𝐽)
2118, 20eqtrid 2788 . . 3 (𝜑 𝑆 = 𝐽)
2221feq3d 6632 . 2 (𝜑 → (𝐹: dom 𝑀 𝑆𝐹: dom 𝑀 𝐽))
2317, 22mpbid 231 1 (𝜑𝐹: dom 𝑀 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3441   cuni 4851  dom cdm 5614  ran crn 5615  wf 6469  cfv 6473  (class class class)co 7329  Basecbs 17001  Scalarcsca 17054   ·𝑠 cvsca 17055  TopOpenctopn 17221  0gc0g 17239  ℝHomcrrh 32182  sigAlgebracsiga 32315  sigaGencsigagen 32345  measurescmeas 32402  sitgcsitg 32537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-1st 7891  df-2nd 7892  df-map 8680  df-esum 32235  df-siga 32316  df-sigagen 32346  df-meas 32403  df-mbfm 32457  df-sitg 32538
This theorem is referenced by:  sibfinima  32547  sibfof  32548  sitgaddlemb  32556  sitmcl  32559
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