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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 37731 | . 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | |
2 | elrelscnveq 37678 | . 2 ⊢ (𝑟 ∈ Rels → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑟 = 𝑟)) | |
3 | 1, 2 | rabimbieq 37435 | 1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 {crab 3431 ⊆ wss 3948 ◡ccnv 5675 Rels crels 37361 SymRels csymrels 37370 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-rels 37671 df-ssr 37684 df-syms 37728 df-symrels 37729 |
This theorem is referenced by: dfsymrels5 37734 elsymrels4 37741 |
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