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 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sibff	⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)

Proof of Theorem sibff

Step	Hyp	Ref	Expression
1		sitgval.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
2		dmmeas 29591	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 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
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V)
8	5, 7	syl5eqel 2692	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
9	8	sgsiga 29532	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
10	4, 9	syl5eqel 2692	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
11		sitgval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
12		sitgval.0	. . . 4 ⊢ 0 = (0g‘𝑊)
13		sitgval.x	. . . 4 ⊢ · = ( ·𝑠 ‘𝑊)
14		sitgval.h	. . . 4 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
15		sitgval.1	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
16		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
17	11, 5, 4, 12, 13, 14, 15, 1, 16	sibfmbl 29724	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
18	3, 10, 17	mbfmf 29644	. 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆)
19	4	unieqi 4381	. . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
20		unisg 29533	. . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
21	8, 20	syl 17	. . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
22	19, 21	syl5eq 2656	. . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
23	22	feq3d 5945	. 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
24	18, 23	mpbid 221	1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)

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  0_gc0g 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