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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elcnvrefrels2 | Structured version Visualization version GIF version |
Description: Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.) |
Ref | Expression |
---|---|
elcnvrefrels2 | ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnvrefrels2 38130 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} | |
2 | id 22 | . . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
3 | dmeq 5906 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
4 | rneq 5938 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅) | |
5 | 3, 4 | xpeq12d 5709 | . . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅)) |
6 | 5 | ineq2d 4210 | . . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅))) |
7 | 2, 6 | sseq12d 4010 | . 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)))) |
8 | 1, 7 | rabeqel 37856 | 1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ⊆ wss 3944 I cid 5575 × cxp 5676 dom cdm 5678 ran crn 5679 Rels crels 37781 CnvRefRels ccnvrefrels 37787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-rels 38087 df-ssr 38100 df-cnvrefs 38127 df-cnvrefrels 38128 |
This theorem is referenced by: elcnvrefrelsrel 38138 cosselcnvrefrels2 38140 |
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