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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 37937 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} | |
2 | id 22 | . . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
3 | dmeq 5900 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
4 | rneq 5932 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅) | |
5 | 3, 4 | xpeq12d 5703 | . . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅)) |
6 | 5 | ineq2d 4208 | . . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅))) |
7 | 2, 6 | sseq12d 4011 | . 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)))) |
8 | 1, 7 | rabeqel 37661 | 1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 I cid 5569 × cxp 5670 dom cdm 5672 ran crn 5673 Rels crels 37585 CnvRefRels ccnvrefrels 37591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-rels 37894 df-ssr 37907 df-cnvrefs 37934 df-cnvrefrels 37935 |
This theorem is referenced by: elcnvrefrelsrel 37945 cosselcnvrefrels2 37947 |
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