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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 40355 | . . 3 ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})) |
8 | 1, 7 | syl5eq 2870 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})) |
9 | breq1 5071 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)) | |
10 | 9 | rabbidv 3482 | . . 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 5236 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V) | |
14 | 4, 13 | syl 17 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V) |
15 | 8, 11, 12, 14 | fvmptd 6777 | 1 ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 𝒫 cpw 4541 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: (None) |
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