![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elrefsymrels2 | Structured version Visualization version GIF version |
Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 37079) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 37004, cf. the comment of dfrefrels2 37004. (Contributed by Peter Mazsa, 22-Jul-2019.) |
Ref | Expression |
---|---|
elrefsymrels2 | ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refsymrels2 37056 | . 2 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} | |
2 | dmeq 5864 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
3 | 2 | reseq2d 5942 | . . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅)) |
4 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
5 | 3, 4 | sseq12d 3982 | . . 3 ⊢ (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
6 | cnveq 5834 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
7 | 6, 4 | sseq12d 3982 | . . 3 ⊢ (𝑟 = 𝑅 → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑅 ⊆ 𝑅)) |
8 | 5, 7 | anbi12d 632 | . 2 ⊢ (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅))) |
9 | 1, 8 | rabeqel 36743 | 1 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3914 ⊆ wss 3915 I cid 5535 ◡ccnv 5637 dom cdm 5638 ↾ cres 5640 Rels crels 36665 RefRels crefrels 36668 SymRels csymrels 36674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-rels 36976 df-ssr 36989 df-refs 37001 df-refrels 37002 df-syms 37033 df-symrels 37034 |
This theorem is referenced by: elrefsymrels3 37061 |
Copyright terms: Public domain | W3C validator |