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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 32800 | . . . 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 6857 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V) | |
7 | 5, 6 | eqeltrid 2842 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) |
8 | 7 | sgsiga 32741 | . . . 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 32935 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) |
17 | 3, 9, 16 | mbfmf 32853 | . 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆) |
18 | 4 | unieqi 4878 | . . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽) |
19 | unisg 32742 | . . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
20 | 7, 19 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
21 | 18, 20 | eqtrid 2788 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
22 | 21 | feq3d 6655 | . 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽)) |
23 | 17, 22 | mpbid 231 | 1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∪ cuni 4865 dom cdm 5633 ran crn 5634 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 Scalarcsca 17136 ·𝑠 cvsca 17137 TopOpenctopn 17303 0gc0g 17321 ℝHomcrrh 32574 sigAlgebracsiga 32707 sigaGencsigagen 32737 measurescmeas 32794 sitgcsitg 32929 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-map 8767 df-esum 32627 df-siga 32708 df-sigagen 32738 df-meas 32795 df-mbfm 32849 df-sitg 32930 |
This theorem is referenced by: sibfinima 32939 sibfof 32940 sitgaddlemb 32948 sitmcl 32951 |
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