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 30395 | . . 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 1074 | 1 ⊢ (𝜑 → ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∖ cdif 3571 {csn 4177 ∪ cuni 4436 ◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 +∞cpnf 10071 [,)cico 12177 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 TopOpenctopn 16082 0gc0g 16100 ℝHomcrrh 30037 sigaGencsigagen 30201 measurescmeas 30258 MblFnMcmbfm 30312 sitgcsitg 30391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-sitg 30392 |
This theorem is referenced by: sibfof 30402 sitgfval 30403 sitgclg 30404 |
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