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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 6076 | . 2 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
2 | relsnopg 5676 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → Rel {〈𝐵, 𝐴〉}) | |
3 | 2 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {〈𝐵, 𝐴〉}) |
4 | dfrel2 6046 | . . 3 ⊢ (Rel {〈𝐵, 𝐴〉} ↔ ◡◡{〈𝐵, 𝐴〉} = {〈𝐵, 𝐴〉}) | |
5 | 3, 4 | sylib 220 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡◡{〈𝐵, 𝐴〉} = {〈𝐵, 𝐴〉}) |
6 | 1, 5 | syl5eqr 2870 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4567 〈cop 4573 ◡ccnv 5554 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 |
This theorem is referenced by: cnvsn 6083 opswap 6086 funsng 6405 f1oprswap 6658 cnvprop 30432 |
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