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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 37034 | . . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 37033 | . . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | 1, 2 | abeqin 36758 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} |
4 | dmexg 7841 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
5 | 4 | elv 3450 | . . . . 5 ⊢ dom 𝑟 ∈ V |
6 | rnexg 7842 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
7 | 6 | elv 3450 | . . . . 5 ⊢ ran 𝑟 ∈ V |
8 | 5, 7 | xpex 7688 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
9 | inex2g 5278 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
10 | brcnvssr 37014 | . . . 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 36823 | . . 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 36756 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3061 {crab 3406 Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5106 I cid 5531 × cxp 5632 ◡ccnv 5633 dom cdm 5634 ran crn 5635 Rels crels 36682 S cssr 36683 CnvRefs ccnvrefs 36687 CnvRefRels ccnvrefrels 36688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-ssr 37006 df-cnvrefs 37033 df-cnvrefrels 37034 |
This theorem is referenced by: elcnvrefrels3 37043 |
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