Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4340 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | sneqd 4137 | . . . 4 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑦〉} = {〈𝐴, 𝑦〉}) |
3 | 2 | cnveqd 5220 | . . 3 ⊢ (𝑥 = 𝐴 → ◡{〈𝑥, 𝑦〉} = ◡{〈𝐴, 𝑦〉}) |
4 | opeq2 4341 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑦, 𝑥〉 = 〈𝑦, 𝐴〉) | |
5 | 4 | sneqd 4137 | . . 3 ⊢ (𝑥 = 𝐴 → {〈𝑦, 𝑥〉} = {〈𝑦, 𝐴〉}) |
6 | 3, 5 | eqeq12d 2625 | . 2 ⊢ (𝑥 = 𝐴 → (◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} ↔ ◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉})) |
7 | opeq2 4341 | . . . . 5 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
8 | 7 | sneqd 4137 | . . . 4 ⊢ (𝑦 = 𝐵 → {〈𝐴, 𝑦〉} = {〈𝐴, 𝐵〉}) |
9 | 8 | cnveqd 5220 | . . 3 ⊢ (𝑦 = 𝐵 → ◡{〈𝐴, 𝑦〉} = ◡{〈𝐴, 𝐵〉}) |
10 | opeq1 4340 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝑦, 𝐴〉 = 〈𝐵, 𝐴〉) | |
11 | 10 | sneqd 4137 | . . 3 ⊢ (𝑦 = 𝐵 → {〈𝑦, 𝐴〉} = {〈𝐵, 𝐴〉}) |
12 | 9, 11 | eqeq12d 2625 | . 2 ⊢ (𝑦 = 𝐵 → (◡{〈𝐴, 𝑦〉} = {〈𝑦, 𝐴〉} ↔ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉})) |
13 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
14 | vex 3176 | . . 3 ⊢ 𝑦 ∈ V | |
15 | 13, 14 | cnvsn 5536 | . 2 ⊢ ◡{〈𝑥, 𝑦〉} = {〈𝑦, 𝑥〉} |
16 | 6, 12, 15 | vtocl2g 3243 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: opswap 5540 funsng 5851 f1oprswap 6092 constr2spthlem1 26124 constr3pthlem2 26184 |
Copyright terms: Public domain | W3C validator |