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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfovfvfvd | 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, relation 𝑅, and left element 𝑋. (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) |
rfovfvd.r | ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) |
rfovfvd.f | ⊢ 𝐹 = (𝐴𝑂𝐵) |
rfovfvfvd.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
rfovfvfvd.g | ⊢ 𝐺 = (𝐹‘𝑅) |
Ref | Expression |
---|---|
rfovfvfvd | ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfovfvfvd.g | . . 3 ⊢ 𝐺 = (𝐹‘𝑅) | |
2 | rfovd.rf | . . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) | |
3 | rfovd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | rfovd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | rfovfvd.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) | |
6 | rfovfvd.f | . . . 4 ⊢ 𝐹 = (𝐴𝑂𝐵) | |
7 | 2, 3, 4, 5, 6 | rfovfvd 40341 | . . 3 ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})) |
8 | 1, 7 | syl5eq 2868 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})) |
9 | breq1 5061 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)) | |
10 | 9 | rabbidv 3480 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦}) |
11 | 10 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦}) |
12 | rfovfvfvd.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
13 | rabexg 5226 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V) | |
14 | 4, 13 | syl 17 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V) |
15 | 8, 11, 12, 14 | fvmptd 6769 | 1 ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 𝒫 cpw 4538 class class class wbr 5058 ↦ cmpt 5138 × cxp 5547 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 |
This theorem is referenced by: (None) |
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