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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrefrels2 | Structured version Visualization version GIF version |
Description: Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.) |
Ref | Expression |
---|---|
elrefrels2 | ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrefrels2 35755 | . 2 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | |
2 | dmeq 5774 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
3 | rneq 5808 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅) | |
4 | 2, 3 | xpeq12d 5588 | . . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅)) |
5 | 4 | ineq2d 4191 | . . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅))) |
6 | id 22 | . . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
7 | 5, 6 | sseq12d 4002 | . 2 ⊢ (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅)) |
8 | 1, 7 | rabeqel 35518 | 1 ⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 I cid 5461 × cxp 5555 dom cdm 5557 ran crn 5558 Rels crels 35457 RefRels crefrels 35460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-rels 35727 df-ssr 35740 df-refs 35752 df-refrels 35753 |
This theorem is referenced by: elrefrelsrel 35761 |
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