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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 all 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 34618 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} | |
2 | id 22 | . . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
3 | dmeq 5462 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
4 | rneq 5489 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅) | |
5 | 3, 4 | xpeq12d 5280 | . . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅)) |
6 | 5 | ineq2d 3965 | . . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅))) |
7 | 2, 6 | sseq12d 3783 | . 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)))) |
8 | 1, 7 | rabeqel 34362 | 1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ⊆ wss 3723 I cid 5156 × cxp 5247 dom cdm 5249 ran crn 5250 Rels crels 34317 CnvRefRels ccnvrefrels 34323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-rels 34577 df-ssr 34590 df-cnvrefs 34615 df-cnvrefrels 34616 |
This theorem is referenced by: elcnvrefrelsrel 34624 cosselcnvrefrels2 34626 |
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