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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 37186 | . 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | |
2 | idinxpss 36984 | . 2 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)) | |
3 | 1, 2 | rabbieq 36921 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∀wral 3060 {crab 3431 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5141 I cid 5566 × cxp 5667 dom cdm 5669 ran crn 5670 Rels crels 36848 RefRels crefrels 36851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-rels 37158 df-ssr 37171 df-refs 37183 df-refrels 37184 |
This theorem is referenced by: elrefrels3 37192 |
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