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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 36243 | . 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | |
2 | idinxpss 36060 | . 2 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)) | |
3 | 1, 2 | rabbieq 36002 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∀wral 3053 {crab 3057 ∩ cin 3840 ⊆ wss 3841 class class class wbr 5027 I cid 5424 × cxp 5517 dom cdm 5519 ran crn 5520 Rels crels 35947 RefRels crefrels 35950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-rels 36215 df-ssr 36228 df-refs 36240 df-refrels 36241 |
This theorem is referenced by: elrefrels3 36248 |
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