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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 36254. (Contributed by Peter Mazsa, 20-Jul-2019.) |
Ref | Expression |
---|---|
dfsymrels2 | ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symrels 36280 | . 2 ⊢ SymRels = ( Syms ∩ Rels ) | |
2 | df-syms 36279 | . 2 ⊢ Syms = {𝑟 ∣ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | inex1g 5187 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
4 | 3 | elv 3404 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
5 | brssr 36242 | . . . 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 36231 | . . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
8 | 7 | elv 3404 | . . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
9 | 8 | biimpi 219 | . . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
10 | 9 | cnveqd 5718 | . . . 4 ⊢ (𝑟 ∈ Rels → ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) = ◡𝑟) |
11 | 10, 9 | sseq12d 3910 | . . 3 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟)) |
12 | 6, 11 | syl5bb 286 | . 2 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟)) |
13 | 1, 2, 12 | abeqinbi 36016 | 1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∈ wcel 2114 {crab 3057 Vcvv 3398 ∩ cin 3842 ⊆ wss 3843 class class class wbr 5030 × cxp 5523 ◡ccnv 5524 dom cdm 5525 ran crn 5526 Rels crels 35958 S cssr 35959 Syms csyms 35966 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-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: dfsymrels3 36283 dfsymrels4 36284 elsymrels2 36290 refsymrels2 36302 refrelsredund4 36368 |
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