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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 37922. (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 37935 | . 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 37934 | . 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | dmexg 7903 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
4 | 3 | elv 3475 | . . . . 5 ⊢ dom 𝑟 ∈ V |
5 | rnexg 7904 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
6 | 5 | elv 3475 | . . . . 5 ⊢ ran 𝑟 ∈ V |
7 | 4, 6 | xpex 7749 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
8 | inex2g 5314 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
9 | brcnvssr 37915 | . . . 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 37899 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
12 | 11 | elv 3475 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
13 | 12 | biimpi 215 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
14 | 13 | sseq1d 4009 | . . 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 37660 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3427 Vcvv 3469 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5142 I cid 5569 × cxp 5670 ◡ccnv 5671 dom cdm 5672 ran crn 5673 Rels crels 37585 S cssr 37586 CnvRefs ccnvrefs 37590 CnvRefRels ccnvrefrels 37591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-rels 37894 df-ssr 37907 df-cnvrefs 37934 df-cnvrefrels 37935 |
This theorem is referenced by: elcnvrefrels2 37943 |
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