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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcnvrefrels2 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 21-Jul-2021.) |
Ref | Expression |
---|---|
dfcnvrefrels2 | ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnvrefrels 35924 | . 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 35923 | . 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | dmexg 7594 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
4 | 3 | elv 3446 | . . . . 5 ⊢ dom 𝑟 ∈ V |
5 | rnexg 7595 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
6 | 5 | elv 3446 | . . . . 5 ⊢ ran 𝑟 ∈ V |
7 | 4, 6 | xpex 7456 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
8 | inex2g 5188 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
9 | brcnvssr 35906 | . . . 4 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))) | |
10 | 7, 8, 9 | mp2b 10 | . . 3 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))) |
11 | elrels6 35890 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
12 | 11 | elv 3446 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
13 | 12 | biimpi 219 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
14 | 13 | sseq1d 3946 | . . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))) |
15 | 10, 14 | syl5bb 286 | . 2 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))) |
16 | 1, 2, 15 | abeqinbi 35675 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 I cid 5424 × cxp 5517 ◡ccnv 5518 dom cdm 5519 ran crn 5520 Rels crels 35615 S cssr 35616 CnvRefs ccnvrefs 35620 CnvRefRels ccnvrefrels 35621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-rels 35885 df-ssr 35898 df-cnvrefs 35923 df-cnvrefrels 35924 |
This theorem is referenced by: elcnvrefrels2 35930 |
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