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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 37907) 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 37883 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
2 | 1 | anbi2d 628 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))) |
3 | elrefrels2 37914 | . 2 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | |
4 | dfrefrel2 37911 | . 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 205 ∧ wa 395 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 I cid 5569 × cxp 5670 dom cdm 5672 ran crn 5673 Rel wrel 5677 Rels crels 37572 RefRels crefrels 37575 RefRel wrefrel 37576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-rels 37881 df-ssr 37894 df-refs 37906 df-refrels 37907 df-refrel 37908 |
This theorem is referenced by: elrefsymrelsrel 37967 |
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