Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibff | Structured version Visualization version GIF version |
Description: A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
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)) |
Ref | Expression |
---|---|
sibff | ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sitgval.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
2 | dmmeas 31464 | . . . 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 6688 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V) | |
7 | 5, 6 | eqeltrid 2920 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) |
8 | 7 | sgsiga 31405 | . . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
9 | 4, 8 | eqeltrid 2920 | . . 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 31597 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) |
17 | 3, 9, 16 | mbfmf 31517 | . 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆) |
18 | 4 | unieqi 4854 | . . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽) |
19 | unisg 31406 | . . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
20 | 7, 19 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
21 | 18, 20 | syl5eq 2871 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
22 | 21 | feq3d 6504 | . 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽)) |
23 | 17, 22 | mpbid 234 | 1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∪ cuni 4841 dom cdm 5558 ran crn 5559 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Scalarcsca 16571 ·𝑠 cvsca 16572 TopOpenctopn 16698 0gc0g 16716 ℝHomcrrh 31238 sigAlgebracsiga 31371 sigaGencsigagen 31401 measurescmeas 31458 sitgcsitg 31591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-map 8411 df-esum 31291 df-siga 31372 df-sigagen 31402 df-meas 31459 df-mbfm 31513 df-sitg 31592 |
This theorem is referenced by: sibfinima 31601 sibfof 31602 sitgaddlemb 31610 sitmcl 31613 |
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