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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 34202 | . . . 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 6921 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑊) ∈ V) | |
| 7 | 5, 6 | eqeltrid 2845 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) | 
| 8 | 7 | sgsiga 34143 | . . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) | 
| 9 | 4, 8 | eqeltrid 2845 | . . 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 34337 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) | 
| 17 | 3, 9, 16 | mbfmf 34255 | . 2 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝑆) | 
| 18 | 4 | unieqi 4919 | . . . 4 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽) | 
| 19 | unisg 34144 | . . . . 5 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
| 20 | 7, 19 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | 
| 21 | 18, 20 | eqtrid 2789 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) | 
| 22 | 21 | feq3d 6723 | . 2 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶∪ 𝑆 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽)) | 
| 23 | 17, 22 | mpbid 232 | 1 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 dom cdm 5685 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 TopOpenctopn 17466 0gc0g 17484 ℝHomcrrh 33994 sigAlgebracsiga 34109 sigaGencsigagen 34139 measurescmeas 34196 sitgcsitg 34331 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-esum 34029 df-siga 34110 df-sigagen 34140 df-meas 34197 df-mbfm 34251 df-sitg 34332 | 
| This theorem is referenced by: sibfinima 34341 sibfof 34342 sitgaddlemb 34350 sitmcl 34353 | 
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