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Mirrors > Home > ILE Home > Th. List > cnvsng | GIF version |
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Ref | Expression |
---|---|
cnvsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3752 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | sneqd 3583 | . . . 4 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑦〉} = {〈𝐴, 𝑦〉}) |
3 | 2 | cnveqd 4774 | . . 3 ⊢ (𝑥 = 𝐴 → ◡{〈𝑥, 𝑦〉} = ◡{〈𝐴, 𝑦〉}) |
4 | opeq2 3753 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑦, 𝑥〉 = 〈𝑦, 𝐴〉) | |
5 | 4 | sneqd 3583 | . . 3 ⊢ (𝑥 = 𝐴 → {〈𝑦, 𝑥〉} = {〈𝑦, 𝐴〉}) |
6 | 3, 5 | eqeq12d 2179 | . 2 ⊢ (𝑥 = 𝐴 → (◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} ↔ ◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉})) |
7 | opeq2 3753 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
8 | 7 | sneqd 3583 | . . . 4 ⊢ (𝑦 = 𝐵 → {〈𝐴, 𝑦〉} = {〈𝐴, 𝐵〉}) |
9 | 8 | cnveqd 4774 | . . 3 ⊢ (𝑦 = 𝐵 → ◡{〈𝐴, 𝑦〉} = ◡{〈𝐴, 𝐵〉}) |
10 | opeq1 3752 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝐴〉 = 〈𝐵, 𝐴〉) | |
11 | 10 | sneqd 3583 | . . 3 ⊢ (𝑦 = 𝐵 → {〈𝑦, 𝐴〉} = {〈𝐵, 𝐴〉}) |
12 | 9, 11 | eqeq12d 2179 | . 2 ⊢ (𝑦 = 𝐵 → (◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉} ↔ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉})) |
13 | vex 2724 | . . 3 ⊢ 𝑥 ∈ V | |
14 | vex 2724 | . . 3 ⊢ 𝑦 ∈ V | |
15 | 13, 14 | cnvsn 5080 | . 2 ⊢ ◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} |
16 | 6, 12, 15 | vtocl2g 2785 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {csn 3570 〈cop 3573 ◡ccnv 4597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 |
This theorem is referenced by: opswapg 5084 funsng 5228 |
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