Theorem sibff 34832

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 34697	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
4		sitgval.s	. . . 4 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
6		fvexd 6897	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V)
7	5, 6	eqeltrid 2866	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34638	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2866	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
10		sitgval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
11		sitgval.0	. . . 4 ⊢ 0 = (0g‘𝑊)
12		sitgval.x	. . . 4 ⊢ · = ( ·𝑠 ‘𝑊)
13		sitgval.h	. . . 4 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
14		sitgval.1	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
15		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
16	10, 5, 4, 11, 12, 13, 14, 1, 15	sibfmbl 34831	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
17	3, 9, 16	mbfmf 34750	. 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆)
18	4	unieqi 4882	. . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
19		unisg 34639	. . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
20	7, 19	syl 18	. . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
21	18, 20	eqtrid 2809	. . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
22	21	feq3d 6691	. 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
23	17, 22	mpbid 235	1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∪ cuni 4870  dom cdm 5659  ran crn 5660  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  Basecbs 17303  Scalarcsca 17347   ·_𝑠 cvsca 17348  TopOpenctopn 17508  0_gc0g 17526  ℝHomcrrh 34488  sigAlgebracsiga 34603  sigaGencsigagen 34634  measurescmeas 34691  sitgcsitg 34825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-map 8831  df-esum 34523  df-siga 34604  df-sigagen 34635  df-meas 34692  df-mbfm 34746  df-sitg 34826

This theorem is used by:  sibfinima  34835  sibfof  34836  sitgaddlemb  34844  sitmcl  34847