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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsymrels2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 20-Jul-2019.) |
| Ref | Expression |
|---|---|
| dfsymrels2 | ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-symrels 38544 | . 2 ⊢ SymRels = ( Syms ∩ Rels ) | |
| 2 | df-syms 38543 | . 2 ⊢ Syms = {𝑟 ∣ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
| 3 | inex1g 5319 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
| 4 | 3 | elv 3485 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
| 5 | brssr 38502 | . . . 4 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)))) | |
| 6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) |
| 7 | elrels6 38491 | . . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
| 8 | 7 | elv 3485 | . . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
| 9 | 8 | biimpi 216 | . . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
| 10 | 9 | cnveqd 5886 | . . . 4 ⊢ (𝑟 ∈ Rels → ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) = ◡𝑟) |
| 11 | 10, 9 | sseq12d 4017 | . . 3 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟)) |
| 12 | 6, 11 | bitrid 283 | . 2 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟)) |
| 13 | 1, 2, 12 | abeqinbi 38254 | 1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 Rels crels 38184 S cssr 38185 Syms csyms 38192 SymRels csymrels 38193 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-rels 38486 df-ssr 38499 df-syms 38543 df-symrels 38544 |
| This theorem is referenced by: dfsymrels3 38547 dfsymrels4 38548 elsymrels2 38554 refsymrels2 38566 refrelsredund4 38633 |
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