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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 34446 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) | 
| 4 | dstfrvel.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) | |
| 5 | 3, 4 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ) | 
| 6 | dstfrvel.2 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) | |
| 7 | breq1 5146 | . . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴)) | |
| 8 | 7 | elrab 3692 | . . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴)) | 
| 9 | 5, 6, 8 | sylanbrc 583 | . . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) | 
| 10 | 3 | ffund 6740 | . . . 4 ⊢ (𝜑 → Fun 𝑋) | 
| 11 | 1, 2 | rrvdm 34448 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) | 
| 12 | 4, 11 | eleqtrrd 2844 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋) | 
| 13 | fvimacnv 7073 | . . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) | |
| 14 | 10, 12, 13 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) | 
| 15 | 9, 14 | mpbid 232 | . 2 ⊢ (𝜑 → 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) | 
| 16 | orvclteel.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 17 | 1, 2, 16 | orrvcval4 34467 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) | 
| 18 | 15, 17 | eleqtrrd 2844 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 {crab 3436 ∪ cuni 4907 class class class wbr 5143 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 ≤ cle 11296 Probcprb 34409 rRndVarcrrv 34442 ∘RV/𝑐corvc 34458 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-top 22900 df-bases 22953 df-esum 34029 df-siga 34110 df-sigagen 34140 df-brsiga 34183 df-meas 34197 df-mbfm 34251 df-prob 34410 df-rrv 34443 df-orvc 34459 | 
| This theorem is referenced by: dstfrvunirn 34477 | 
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