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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcnvssr | Structured version Visualization version GIF version |
Description: The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.) |
Ref | Expression |
---|---|
brcnvssr | ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relssr 34791 | . . 3 ⊢ Rel S | |
2 | 1 | relbrcnv 5747 | . 2 ⊢ (𝐴◡ S 𝐵 ↔ 𝐵 S 𝐴) |
3 | brssr 34792 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 S 𝐴 ↔ 𝐵 ⊆ 𝐴)) | |
4 | 2, 3 | syl5bb 275 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 ⊆ wss 3798 class class class wbr 4873 ◡ccnv 5341 S cssr 34520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-ssr 34789 |
This theorem is referenced by: brcnvssrid 34798 br1cossxrncnvssrres 34799 dfcnvrefrels2 34817 dfcnvrefrels3 34818 |
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