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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcnvrefrels3 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the class of converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.) |
| Ref | Expression |
|---|---|
| dfcnvrefrels3 | ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-cnvrefrels 38527 | . . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
| 2 | df-cnvrefs 38526 | . . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
| 3 | 1, 2 | abeqin 38253 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} |
| 4 | dmexg 7923 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
| 5 | 4 | elv 3485 | . . . . 5 ⊢ dom 𝑟 ∈ V |
| 6 | rnexg 7924 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
| 7 | 6 | elv 3485 | . . . . 5 ⊢ ran 𝑟 ∈ V |
| 8 | 5, 7 | xpex 7773 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
| 9 | inex2g 5320 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
| 10 | brcnvssr 38507 | . . . 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)))) | |
| 11 | 8, 9, 10 | mp2b 10 | . . 3 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))) |
| 12 | inxpssidinxp 38317 | . . 3 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)) | |
| 13 | 11, 12 | bitri 275 | . 2 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)) |
| 14 | 3, 13 | rabbieq 3445 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 I cid 5577 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 Rels crels 38184 S cssr 38185 CnvRefs ccnvrefs 38189 CnvRefRels ccnvrefrels 38190 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-ssr 38499 df-cnvrefs 38526 df-cnvrefrels 38527 |
| This theorem is referenced by: elcnvrefrels3 38536 |
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