Theorem sibff 34520

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 34385	. . . 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		fvexd 6859	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V)
7	5, 6	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34326	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2841	. . 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 34519	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
17	3, 9, 16	mbfmf 34438	. 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆)
18	4	unieqi 4877	. . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
19		unisg 34327	. . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
20	7, 19	syl 17	. . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
21	18, 20	eqtrid 2784	. . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
22	21	feq3d 6657	. 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
23	17, 22	mpbid 232	1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∪ cuni 4865  dom cdm 5634  ran crn 5635  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  Scalarcsca 17194   ·_𝑠 cvsca 17195  TopOpenctopn 17355  0_gc0g 17373  ℝHomcrrh 34177  sigAlgebracsiga 34292  sigaGencsigagen 34322  measurescmeas 34379  sitgcsitg 34513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-map 8779  df-esum 34212  df-siga 34293  df-sigagen 34323  df-meas 34380  df-mbfm 34434  df-sitg 34514

This theorem is referenced by:  sibfinima  34523  sibfof  34524  sitgaddlemb  34532  sitmcl  34535