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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.) (Proof shortened by BJ, 12-Feb-2022.) |
Ref | Expression |
---|---|
cnvsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnvsn 6225 | . 2 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
2 | relsnopg 5805 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → Rel {〈𝐵, 𝐴〉}) | |
3 | 2 | ancoms 457 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {〈𝐵, 𝐴〉}) |
4 | dfrel2 6195 | . . 3 ⊢ (Rel {〈𝐵, 𝐴〉} ↔ ◡◡{〈𝐵, 𝐴〉} = {〈𝐵, 𝐴〉}) | |
5 | 3, 4 | sylib 217 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡◡{〈𝐵, 𝐴〉} = {〈𝐵, 𝐴〉}) |
6 | 1, 5 | eqtr3id 2779 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4630 〈cop 4636 ◡ccnv 5677 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 |
This theorem is referenced by: cnvsn 6232 opswap 6235 funsng 6605 f1oprswap 6882 cnvprop 32558 |
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