Theorem sibff 34495

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 34360	. . . 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 6850	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V)
7	5, 6	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34301	. . . 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 34494	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
17	3, 9, 16	mbfmf 34413	. 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆)
18	4	unieqi 4876	. . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
19		unisg 34302	. . . . 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 6648	. 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 3441  ∪ cuni 4864  dom cdm 5625  ran crn 5626  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  Scalarcsca 17184   ·_𝑠 cvsca 17185  TopOpenctopn 17345  0_gc0g 17363  ℝHomcrrh 34152  sigAlgebracsiga 34267  sigaGencsigagen 34297  measurescmeas 34354  sitgcsitg 34488

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-esum 34187  df-siga 34268  df-sigagen 34298  df-meas 34355  df-mbfm 34409  df-sitg 34489

This theorem is referenced by:  sibfinima  34498  sibfof  34499  sitgaddlemb  34507  sitmcl  34510