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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfrn | Structured version Visualization version GIF version |
Description: A simple function has finite range. (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 |
---|---|
sibfrn | ⊢ (𝜑 → ran 𝐹 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sibfmbl.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) | |
2 | sitgval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | sitgval.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | sitgval.s | . . . 4 ⊢ 𝑆 = (sigaGen‘𝐽) | |
5 | sitgval.0 | . . . 4 ⊢ 0 = (0g‘𝑊) | |
6 | sitgval.x | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | sitgval.h | . . . 4 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) | |
8 | sitgval.1 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑉) | |
9 | sitgval.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | issibf 34084 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))) |
11 | 1, 10 | mpbid 231 | . 2 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))) |
12 | 11 | simp2d 1140 | 1 ⊢ (𝜑 → ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∖ cdif 3941 {csn 4630 ∪ cuni 4909 ◡ccnv 5677 dom cdm 5678 ran crn 5679 “ cima 5681 ‘cfv 6549 (class class class)co 7419 Fincfn 8964 0cc0 11140 +∞cpnf 11277 [,)cico 13361 Basecbs 17183 Scalarcsca 17239 ·𝑠 cvsca 17240 TopOpenctopn 17406 0gc0g 17424 ℝHomcrrh 33725 sigaGencsigagen 33888 measurescmeas 33945 MblFnMcmbfm 33999 sitgcsitg 34080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-sitg 34081 |
This theorem is referenced by: sibfof 34091 sitgfval 34092 sitgclg 34093 |
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