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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrefrelsrel | Structured version Visualization version GIF version |
Description: For sets, being an element of the class of reflexive relations (df-refrels 38493) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
Ref | Expression |
---|---|
elrefrelsrel | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrelsrel 38469 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
2 | 1 | anbi2d 630 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))) |
3 | elrefrels2 38500 | . 2 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | |
4 | dfrefrel2 38497 | . 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)) | |
5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 I cid 5582 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 Rels crels 38164 RefRels crefrels 38167 RefRel wrefrel 38168 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-rels 38467 df-ssr 38480 df-refs 38492 df-refrels 38493 df-refrel 38494 |
This theorem is referenced by: elrefsymrelsrel 38553 |
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