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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 34459 | . . 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 482 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑥 = 𝑋) | |
| 4 | 3 | cnveqd 5886 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → ◡𝑥 = ◡𝑋) |
| 5 | simpr 484 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴) | |
| 6 | 5 | breq2d 5155 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (𝑦𝑅𝑎 ↔ 𝑦𝑅𝐴)) |
| 7 | 6 | abbidv 2808 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → {𝑦 ∣ 𝑦𝑅𝑎} = {𝑦 ∣ 𝑦𝑅𝐴}) |
| 8 | 4, 7 | imaeq12d 6079 | . . 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.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 11 | orvcval.1 | . . 3 ⊢ (𝜑 → Fun 𝑋) | |
| 12 | funeq 6586 | . . 3 ⊢ (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋)) | |
| 13 | 10, 11, 12 | elabd 3681 | . 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∣ Fun 𝑥}) |
| 14 | orvcval.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
| 15 | elex 3501 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
| 16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
| 17 | cnvexg 7946 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ◡𝑋 ∈ V) | |
| 18 | imaexg 7935 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V) | |
| 19 | 10, 17, 18 | 3syl 18 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V) |
| 20 | 2, 9, 13, 16, 19 | ovmpod 7585 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 class class class wbr 5143 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 (class class class)co 7431 ∈ cmpo 7433 ∘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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-br 5144 df-opab 5206 df-id 5578 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-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-orvc 34459 |
| This theorem is referenced by: orvcval2 34461 orvcval4 34463 |
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