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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrefrels3 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of reflexive relations. (Contributed by Peter Mazsa, 8-Jul-2019.) |
Ref | Expression |
---|---|
dfrefrels3 | ⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrefrels2 38469 | . 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | |
2 | idinxpss 38268 | . 2 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)) | |
3 | 1, 2 | rabbieq 3452 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∀wral 3067 {crab 3443 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 I cid 5592 × cxp 5698 dom cdm 5700 ran crn 5701 Rels crels 38137 RefRels crefrels 38140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-dm 5710 df-rn 5711 df-res 5712 df-rels 38441 df-ssr 38454 df-refs 38466 df-refrels 38467 |
This theorem is referenced by: elrefrels3 38475 |
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