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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 35758 | . . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 35757 | . . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | 1, 2 | abeqin 35508 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} |
4 | dmexg 7607 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
5 | 4 | elv 3499 | . . . . 5 ⊢ dom 𝑟 ∈ V |
6 | rnexg 7608 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
7 | 6 | elv 3499 | . . . . 5 ⊢ ran 𝑟 ∈ V |
8 | 5, 7 | xpex 7470 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
9 | inex2g 5216 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
10 | brcnvssr 35740 | . . . 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 35567 | . . 3 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)) | |
13 | 11, 12 | bitri 277 | . 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 35506 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 I cid 5453 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 Rels crels 35449 S cssr 35450 CnvRefs ccnvrefs 35454 CnvRefRels ccnvrefrels 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-ssr 35732 df-cnvrefs 35757 df-cnvrefrels 35758 |
This theorem is referenced by: elcnvrefrels3 35765 |
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