Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dstfrvel | Structured version Visualization version GIF version |
Description: Elementhood of preimage maps produced by the "lower than or equal" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.) |
Ref | Expression |
---|---|
dstfrv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
dstfrv.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orvclteel.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dstfrvel.1 | ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) |
dstfrvel.2 | ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) |
Ref | Expression |
---|---|
dstfrvel | ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | dstfrv.2 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | 1, 2 | rrvvf 30506 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
4 | dstfrvel.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) | |
5 | 3, 4 | ffvelrnd 6360 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ) |
6 | dstfrvel.2 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) | |
7 | breq1 4656 | . . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴)) | |
8 | 7 | elrab 3363 | . . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴)) |
9 | 5, 6, 8 | sylanbrc 698 | . . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) |
10 | ffun 6048 | . . . . 5 ⊢ (𝑋:∪ dom 𝑃⟶ℝ → Fun 𝑋) | |
11 | 3, 10 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝑋) |
12 | 1, 2 | rrvdm 30508 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
13 | 4, 12 | eleqtrrd 2704 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋) |
14 | fvimacnv 6332 | . . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) | |
15 | 11, 13, 14 | syl2anc 693 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) |
16 | 9, 15 | mpbid 222 | . 2 ⊢ (𝜑 → 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
17 | orvclteel.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
18 | 1, 2, 17 | orrvcval4 30526 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
19 | 16, 18 | eleqtrrd 2704 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 {crab 2916 ∪ cuni 4436 class class class wbr 4653 ◡ccnv 5113 dom cdm 5114 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 ≤ cle 10075 Probcprb 30469 rRndVarcrrv 30502 ∘RV/𝑐corvc 30517 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-nel 2898 df-ral 2917 df-rex 2918 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-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-topgen 16104 df-top 20699 df-bases 20750 df-esum 30090 df-siga 30171 df-sigagen 30202 df-brsiga 30245 df-meas 30259 df-mbfm 30313 df-prob 30470 df-rrv 30503 df-orvc 30518 |
This theorem is referenced by: dstfrvunirn 30536 |
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