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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 35747. (Contributed by Peter Mazsa, 20-Jul-2019.) |
Ref | Expression |
---|---|
dfsymrels2 | ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symrels 35773 | . 2 ⊢ SymRels = ( Syms ∩ Rels ) | |
2 | df-syms 35772 | . 2 ⊢ Syms = {𝑟 ∣ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | inex1g 5215 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
4 | 3 | elv 3499 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
5 | brssr 35735 | . . . 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 35724 | . . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
8 | 7 | elv 3499 | . . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
9 | 8 | biimpi 218 | . . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
10 | 9 | cnveqd 5740 | . . . 4 ⊢ (𝑟 ∈ Rels → ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) = ◡𝑟) |
11 | 10, 9 | sseq12d 3999 | . . 3 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟)) |
12 | 6, 11 | syl5bb 285 | . 2 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟)) |
13 | 1, 2, 12 | abeqinbi 35509 | 1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 Rels crels 35449 S cssr 35450 Syms csyms 35457 SymRels csymrels 35458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-rels 35719 df-ssr 35732 df-syms 35772 df-symrels 35773 |
This theorem is referenced by: dfsymrels3 35776 dfsymrels4 35777 elsymrels2 35783 refsymrels2 35795 refrelsredund4 35861 |
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