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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfovd | Structured version Visualization version GIF version |
Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.) |
Ref | Expression |
---|---|
rfovd.rf | ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) |
rfovd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
rfovd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
rfovd | ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfovd.rf | . . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))) |
3 | xpeq12 5692 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵)) | |
4 | 3 | pweqd 4612 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵)) |
5 | simpl 482 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴) | |
6 | rabeq 3438 | . . . . . 6 ⊢ (𝑏 = 𝐵 → {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) | |
7 | 6 | adantl 481 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
8 | 5, 7 | mpteq12dv 5230 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
9 | 4, 8 | mpteq12dv 5230 | . . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
10 | 9 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
11 | rfovd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | 11 | elexd 3487 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
13 | rfovd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
14 | 13 | elexd 3487 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
15 | 11, 13 | xpexd 7732 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
16 | pwexg 5367 | . . 3 ⊢ ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V) | |
17 | mptexg 7215 | . . 3 ⊢ (𝒫 (𝐴 × 𝐵) ∈ V → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∈ V) | |
18 | 15, 16, 17 | 3syl 18 | . 2 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∈ V) |
19 | 2, 10, 12, 14, 18 | ovmpod 7553 | 1 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 𝒫 cpw 4595 class class class wbr 5139 ↦ cmpt 5222 × cxp 5665 (class class class)co 7402 ∈ cmpo 7404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 |
This theorem is referenced by: rfovfvd 43303 rfovcnvf1od 43305 fsovrfovd 43310 |
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