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 29591 | . . . 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 | fvex 6113 | . . . . . . 7 ⊢ (TopOpen‘𝑊) ∈ V | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V) |
8 | 5, 7 | syl5eqel 2692 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) |
9 | 8 | sgsiga 29532 | . . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
10 | 4, 9 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
11 | sitgval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
12 | sitgval.0 | . . . 4 ⊢ 0 = (0g‘𝑊) | |
13 | sitgval.x | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
14 | sitgval.h | . . . 4 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) | |
15 | sitgval.1 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑉) | |
16 | sibfmbl.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) | |
17 | 11, 5, 4, 12, 13, 14, 15, 1, 16 | sibfmbl 29724 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) |
18 | 3, 10, 17 | mbfmf 29644 | . 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆) |
19 | 4 | unieqi 4381 | . . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽) |
20 | unisg 29533 | . . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
21 | 8, 20 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
22 | 19, 21 | syl5eq 2656 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
23 | 22 | feq3d 5945 | . 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽)) |
24 | 18, 23 | mpbid 221 | 1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 dom cdm 5038 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 TopOpenctopn 15905 0gc0g 15923 ℝHomcrrh 29365 sigAlgebracsiga 29497 sigaGencsigagen 29528 measurescmeas 29585 sitgcsitg 29718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-esum 29417 df-siga 29498 df-sigagen 29529 df-meas 29586 df-mbfm 29640 df-sitg 29719 |
This theorem is referenced by: sibfinima 29728 sibfof 29729 sitgaddlemb 29737 sitmcl 29740 |
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