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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvelrels | Structured version Visualization version GIF version |
Description: The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.) |
Ref | Expression |
---|---|
cnvelrels | ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5972 | . 2 ⊢ Rel ◡𝐴 | |
2 | cnvexg 7702 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
3 | elrelsrel 36342 | . . 3 ⊢ (◡𝐴 ∈ V → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (◡𝐴 ∈ Rels ↔ Rel ◡𝐴)) |
5 | 1, 4 | mpbiri 261 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2110 Vcvv 3408 ◡ccnv 5550 Rel wrel 5556 Rels crels 36072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-rels 36340 |
This theorem is referenced by: cosscnvelrels 36352 |
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