Theorem sibff 34349

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 34214	. . . 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 6837	. . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V)
7	5, 6	eqeltrid 2835	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34155	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2835	. . 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 34348	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
17	3, 9, 16	mbfmf 34267	. 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆)
18	4	unieqi 4868	. . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
19		unisg 34156	. . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
20	7, 19	syl 17	. . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
21	18, 20	eqtrid 2778	. . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
22	21	feq3d 6636	. 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 1541   ∈ wcel 2111  Vcvv 3436  ∪ cuni 4856  dom cdm 5614  ran crn 5615  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Scalarcsca 17164   ·_𝑠 cvsca 17165  TopOpenctopn 17325  0_gc0g 17343  ℝHomcrrh 34006  sigAlgebracsiga 34121  sigaGencsigagen 34151  measurescmeas 34208  sitgcsitg 34342

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  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 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-esum 34041  df-siga 34122  df-sigagen 34152  df-meas 34209  df-mbfm 34263  df-sitg 34343

This theorem is referenced by:  sibfinima  34352  sibfof  34353  sitgaddlemb  34361  sitmcl  34364