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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 3705 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | sneqd 3540 | . . . 4 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑦〉} = {〈𝐴, 𝑦〉}) |
3 | 2 | cnveqd 4715 | . . 3 ⊢ (𝑥 = 𝐴 → ◡{〈𝑥, 𝑦〉} = ◡{〈𝐴, 𝑦〉}) |
4 | opeq2 3706 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑦, 𝑥〉 = 〈𝑦, 𝐴〉) | |
5 | 4 | sneqd 3540 | . . 3 ⊢ (𝑥 = 𝐴 → {〈𝑦, 𝑥〉} = {〈𝑦, 𝐴〉}) |
6 | 3, 5 | eqeq12d 2154 | . 2 ⊢ (𝑥 = 𝐴 → (◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} ↔ ◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉})) |
7 | opeq2 3706 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
8 | 7 | sneqd 3540 | . . . 4 ⊢ (𝑦 = 𝐵 → {〈𝐴, 𝑦〉} = {〈𝐴, 𝐵〉}) |
9 | 8 | cnveqd 4715 | . . 3 ⊢ (𝑦 = 𝐵 → ◡{〈𝐴, 𝑦〉} = ◡{〈𝐴, 𝐵〉}) |
10 | opeq1 3705 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝐴〉 = 〈𝐵, 𝐴〉) | |
11 | 10 | sneqd 3540 | . . 3 ⊢ (𝑦 = 𝐵 → {〈𝑦, 𝐴〉} = {〈𝐵, 𝐴〉}) |
12 | 9, 11 | eqeq12d 2154 | . 2 ⊢ (𝑦 = 𝐵 → (◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉} ↔ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉})) |
13 | vex 2689 | . . 3 ⊢ 𝑥 ∈ V | |
14 | vex 2689 | . . 3 ⊢ 𝑦 ∈ V | |
15 | 13, 14 | cnvsn 5021 | . 2 ⊢ ◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} |
16 | 6, 12, 15 | vtocl2g 2750 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3527 〈cop 3530 ◡ccnv 4538 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 |
This theorem is referenced by: opswapg 5025 funsng 5169 |
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