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 31881 | . . . 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 6732 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V) | |
7 | 5, 6 | eqeltrid 2842 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) |
8 | 7 | sgsiga 31822 | . . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
9 | 4, 8 | eqeltrid 2842 | . . 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 32014 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) |
17 | 3, 9, 16 | mbfmf 31934 | . 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆) |
18 | 4 | unieqi 4832 | . . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽) |
19 | unisg 31823 | . . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
20 | 7, 19 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
21 | 18, 20 | syl5eq 2790 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
22 | 21 | feq3d 6532 | . 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽)) |
23 | 17, 22 | mpbid 235 | 1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∪ cuni 4819 dom cdm 5551 ran crn 5552 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 Scalarcsca 16805 ·𝑠 cvsca 16806 TopOpenctopn 16926 0gc0g 16944 ℝHomcrrh 31655 sigAlgebracsiga 31788 sigaGencsigagen 31818 measurescmeas 31875 sitgcsitg 32008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-map 8510 df-esum 31708 df-siga 31789 df-sigagen 31819 df-meas 31876 df-mbfm 31930 df-sitg 32009 |
This theorem is referenced by: sibfinima 32018 sibfof 32019 sitgaddlemb 32027 sitmcl 32030 |
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