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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfovcnvd | Structured version Visualization version GIF version |
Description: Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-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) |
rfovcnvf1od.f | ⊢ 𝐹 = (𝐴𝑂𝐵) |
Ref | Expression |
---|---|
rfovcnvd | ⊢ (𝜑 → ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfovd.rf | . . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) | |
2 | rfovd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | rfovd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | rfovcnvf1od.f | . . 3 ⊢ 𝐹 = (𝐴𝑂𝐵) | |
5 | 1, 2, 3, 4 | rfovcnvf1od 38298 | . 2 ⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))) |
6 | 5 | simprd 479 | 1 ⊢ (𝜑 → ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 𝒫 cpw 4158 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 |
This theorem is referenced by: rfovcnvfvd 38301 fsovrfovd 38303 |
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