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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 35633. (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 35644 | . 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 35643 | . 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | dmexg 7602 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
4 | 3 | elv 3497 | . . . . 5 ⊢ dom 𝑟 ∈ V |
5 | rnexg 7603 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
6 | 5 | elv 3497 | . . . . 5 ⊢ ran 𝑟 ∈ V |
7 | 4, 6 | xpex 7465 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
8 | inex2g 5215 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
9 | brcnvssr 35626 | . . . 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 35610 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
12 | 11 | elv 3497 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
13 | 12 | biimpi 217 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
14 | 13 | sseq1d 3995 | . . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))) |
15 | 10, 14 | syl5bb 284 | . 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 35396 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 I cid 5452 × cxp 5546 ◡ccnv 5547 dom cdm 5548 ran crn 5549 Rels crels 35336 S cssr 35337 CnvRefs ccnvrefs 35341 CnvRefRels ccnvrefrels 35342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-rels 35605 df-ssr 35618 df-cnvrefs 35643 df-cnvrefrels 35644 |
This theorem is referenced by: elcnvrefrels2 35650 |
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