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Mirrors > Home > MPE Home > Th. List > cnvsng | Structured version Visualization version 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 4402 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | sneqd 4189 | . . . 4 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑦〉} = {〈𝐴, 𝑦〉}) |
3 | 2 | cnveqd 5298 | . . 3 ⊢ (𝑥 = 𝐴 → ◡{〈𝑥, 𝑦〉} = ◡{〈𝐴, 𝑦〉}) |
4 | opeq2 4403 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑦, 𝑥〉 = 〈𝑦, 𝐴〉) | |
5 | 4 | sneqd 4189 | . . 3 ⊢ (𝑥 = 𝐴 → {〈𝑦, 𝑥〉} = {〈𝑦, 𝐴〉}) |
6 | 3, 5 | eqeq12d 2637 | . 2 ⊢ (𝑥 = 𝐴 → (◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} ↔ ◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉})) |
7 | opeq2 4403 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
8 | 7 | sneqd 4189 | . . . 4 ⊢ (𝑦 = 𝐵 → {〈𝐴, 𝑦〉} = {〈𝐴, 𝐵〉}) |
9 | 8 | cnveqd 5298 | . . 3 ⊢ (𝑦 = 𝐵 → ◡{〈𝐴, 𝑦〉} = ◡{〈𝐴, 𝐵〉}) |
10 | opeq1 4402 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝐴〉 = 〈𝐵, 𝐴〉) | |
11 | 10 | sneqd 4189 | . . 3 ⊢ (𝑦 = 𝐵 → {〈𝑦, 𝐴〉} = {〈𝐵, 𝐴〉}) |
12 | 9, 11 | eqeq12d 2637 | . 2 ⊢ (𝑦 = 𝐵 → (◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉} ↔ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉})) |
13 | vex 3203 | . . 3 ⊢ 𝑥 ∈ V | |
14 | vex 3203 | . . 3 ⊢ 𝑦 ∈ V | |
15 | 13, 14 | cnvsn 5618 | . 2 ⊢ ◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} |
16 | 6, 12, 15 | vtocl2g 3270 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 〈cop 4183 ◡ccnv 5113 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: opswap 5622 funsng 5937 f1oprswap 6180 |
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