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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 38469. (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 38482 | . 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 38481 | . 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | dmexg 7941 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
4 | 3 | elv 3493 | . . . . 5 ⊢ dom 𝑟 ∈ V |
5 | rnexg 7942 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
6 | 5 | elv 3493 | . . . . 5 ⊢ ran 𝑟 ∈ V |
7 | 4, 6 | xpex 7788 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
8 | inex2g 5338 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
9 | brcnvssr 38462 | . . . 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 38446 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
12 | 11 | elv 3493 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
13 | 12 | biimpi 216 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
14 | 13 | sseq1d 4040 | . . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))) |
15 | 10, 14 | bitrid 283 | . 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 38209 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2108 {crab 3443 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 I cid 5592 × cxp 5698 ◡ccnv 5699 dom cdm 5700 ran crn 5701 Rels crels 38137 S cssr 38138 CnvRefs ccnvrefs 38142 CnvRefRels ccnvrefrels 38143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-dm 5710 df-rn 5711 df-res 5712 df-rels 38441 df-ssr 38454 df-cnvrefs 38481 df-cnvrefrels 38482 |
This theorem is referenced by: elcnvrefrels2 38490 |
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