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 "less than or equal to" 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 31702 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
4 | dstfrvel.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) | |
5 | 3, 4 | ffvelrnd 6852 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ) |
6 | dstfrvel.2 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) | |
7 | breq1 5069 | . . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴)) | |
8 | 7 | elrab 3680 | . . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴)) |
9 | 5, 6, 8 | sylanbrc 585 | . . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) |
10 | 3 | ffund 6518 | . . . 4 ⊢ (𝜑 → Fun 𝑋) |
11 | 1, 2 | rrvdm 31704 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
12 | 4, 11 | eleqtrrd 2916 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋) |
13 | fvimacnv 6823 | . . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) | |
14 | 10, 12, 13 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) |
15 | 9, 14 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
16 | orvclteel.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
17 | 1, 2, 16 | orrvcval4 31722 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
18 | 15, 17 | eleqtrrd 2916 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 {crab 3142 ∪ cuni 4838 class class class wbr 5066 ◡ccnv 5554 dom cdm 5555 “ cima 5558 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 ≤ cle 10676 Probcprb 31665 rRndVarcrrv 31698 ∘RV/𝑐corvc 31713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 df-topgen 16717 df-top 21502 df-bases 21554 df-esum 31287 df-siga 31368 df-sigagen 31398 df-brsiga 31441 df-meas 31455 df-mbfm 31509 df-prob 31666 df-rrv 31699 df-orvc 31714 |
This theorem is referenced by: dstfrvunirn 31732 |
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