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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 30523 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))) |
11 | 1, 10 | mpbid 222 | . 2 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))) |
12 | 11 | simp2d 1094 | 1 ⊢ (𝜑 → ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∖ cdif 3604 {csn 4210 ∪ cuni 4468 ◡ccnv 5142 dom cdm 5143 ran crn 5144 “ cima 5146 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 0cc0 9974 +∞cpnf 10109 [,)cico 12215 Basecbs 15904 Scalarcsca 15991 ·𝑠 cvsca 15992 TopOpenctopn 16129 0gc0g 16147 ℝHomcrrh 30165 sigaGencsigagen 30329 measurescmeas 30386 MblFnMcmbfm 30440 sitgcsitg 30519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-sitg 30520 |
This theorem is referenced by: sibfof 30530 sitgfval 30531 sitgclg 30532 |
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