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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsymrels4 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.) |
Ref | Expression |
---|---|
dfsymrels4 | ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsymrels2 36282 | . 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | |
2 | elrelscnveq 36233 | . 2 ⊢ (𝑟 ∈ Rels → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑟 = 𝑟)) | |
3 | 1, 2 | rabimbieq 36014 | 1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {crab 3057 ⊆ wss 3843 ◡ccnv 5524 Rels crels 35958 SymRels csymrels 35967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-xp 5531 df-rel 5532 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-rels 36226 df-ssr 36239 df-syms 36279 df-symrels 36280 |
This theorem is referenced by: dfsymrels5 36285 elsymrels4 36292 |
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