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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 37036 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} | |
2 | id 22 | . . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
3 | dmeq 5860 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
4 | rneq 5892 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅) | |
5 | 3, 4 | xpeq12d 5665 | . . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅)) |
6 | 5 | ineq2d 4173 | . . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅))) |
7 | 2, 6 | sseq12d 3978 | . 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)))) |
8 | 1, 7 | rabeqel 36760 | 1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 I cid 5531 × cxp 5632 dom cdm 5634 ran crn 5635 Rels crels 36682 CnvRefRels ccnvrefrels 36688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-rels 36993 df-ssr 37006 df-cnvrefs 37033 df-cnvrefrels 37034 |
This theorem is referenced by: elcnvrefrelsrel 37044 cosselcnvrefrels2 37046 |
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