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 29833 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
4 | dstfrvel.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) | |
5 | 3, 4 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ) |
6 | dstfrvel.2 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) | |
7 | breq1 4586 | . . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴)) | |
8 | 7 | elrab 3331 | . . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴)) |
9 | 5, 6, 8 | sylanbrc 695 | . . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) |
10 | ffun 5961 | . . . . 5 ⊢ (𝑋:∪ dom 𝑃⟶ℝ → Fun 𝑋) | |
11 | 3, 10 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝑋) |
12 | 1, 2 | rrvdm 29835 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
13 | 4, 12 | eleqtrrd 2691 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋) |
14 | fvimacnv 6240 | . . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) | |
15 | 11, 13, 14 | syl2anc 691 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) |
16 | 9, 15 | mpbid 221 | . 2 ⊢ (𝜑 → 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
17 | orvclteel.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
18 | 1, 2, 17 | orrvcval4 29853 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
19 | 16, 18 | eleqtrrd 2691 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 {crab 2900 ∪ cuni 4372 class class class wbr 4583 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ≤ cle 9954 Probcprb 29796 rRndVarcrrv 29829 ∘RV/𝑐corvc 29844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-topgen 15927 df-top 20521 df-bases 20522 df-esum 29417 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-meas 29586 df-mbfm 29640 df-prob 29797 df-rrv 29830 df-orvc 29845 |
This theorem is referenced by: dstfrvunirn 29863 |
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