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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elcnvrefrelsrel | Structured version Visualization version GIF version | ||
| Description: For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38527) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
| Ref | Expression |
|---|---|
| elcnvrefrelsrel | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elrelsrel 38488 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
| 2 | 1 | anbi2d 630 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))) |
| 3 | elcnvrefrels2 38535 | . 2 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) | |
| 4 | dfcnvrefrel2 38531 | . 2 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | |
| 5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 I cid 5577 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 Rels crels 38184 CnvRefRels ccnvrefrels 38190 CnvRefRel wcnvrefrel 38191 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-rels 38486 df-ssr 38499 df-cnvrefs 38526 df-cnvrefrels 38527 df-cnvrefrel 38528 |
| This theorem is referenced by: elfunsALTVfunALTV 38698 eldisjsdisj 38728 |
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