Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcval | Structured version Visualization version GIF version |
Description: Value of the preimage mapping operator applied on a given random variable and constant. (Contributed by Thierry Arnoux, 19-Jan-2017.) |
Ref | Expression |
---|---|
orvcval.1 | ⊢ (𝜑 → Fun 𝑋) |
orvcval.2 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
orvcval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
Ref | Expression |
---|---|
orvcval | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-orvc 29845 | . . 3 ⊢ ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))) |
3 | simpl 472 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑥 = 𝑋) | |
4 | 3 | cnveqd 5220 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → ◡𝑥 = ◡𝑋) |
5 | simpr 476 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴) | |
6 | 5 | breq2d 4595 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (𝑦𝑅𝑎 ↔ 𝑦𝑅𝐴)) |
7 | 6 | abbidv 2728 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → {𝑦 ∣ 𝑦𝑅𝑎} = {𝑦 ∣ 𝑦𝑅𝐴}) |
8 | 4, 7 | imaeq12d 5386 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
9 | 8 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑎 = 𝐴)) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
10 | orvcval.1 | . . 3 ⊢ (𝜑 → Fun 𝑋) | |
11 | orvcval.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
12 | funeq 5823 | . . . . 5 ⊢ (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋)) | |
13 | 12 | elabg 3320 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ Fun 𝑥} ↔ Fun 𝑋)) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝑋 ∈ {𝑥 ∣ Fun 𝑥} ↔ Fun 𝑋)) |
15 | 10, 14 | mpbird 246 | . 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∣ Fun 𝑥}) |
16 | orvcval.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
17 | elex 3185 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
19 | cnvexg 7005 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ◡𝑋 ∈ V) | |
20 | imaexg 6995 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V) | |
21 | 11, 19, 20 | 3syl 18 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V) |
22 | 2, 9, 15, 18, 21 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 Fun wfun 5798 (class class class)co 6549 ↦ cmpt2 6551 ∘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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-br 4584 df-opab 4644 df-id 4953 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-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-orvc 29845 |
This theorem is referenced by: orvcval2 29847 orvcval4 29849 |
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