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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 34902 | . . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 34901 | . . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | 1, 2 | abeqin 34652 | . 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} |
4 | dmexg 7375 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
5 | 4 | elv 3402 | . . . . 5 ⊢ dom 𝑟 ∈ V |
6 | rnexg 7376 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
7 | 6 | elv 3402 | . . . . 5 ⊢ ran 𝑟 ∈ V |
8 | 5, 7 | xpex 7240 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
9 | inex2ALTV 34733 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
10 | brcnvssr 34884 | . . . 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 34715 | . . 3 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)) | |
13 | 11, 12 | bitri 267 | . 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 34650 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∀wral 3090 {crab 3094 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 class class class wbr 4886 I cid 5260 × cxp 5353 ◡ccnv 5354 dom cdm 5355 ran crn 5356 Rels crels 34608 S cssr 34609 CnvRefs ccnvrefs 34613 CnvRefRels ccnvrefrels 34614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-dm 5365 df-rn 5366 df-ssr 34876 df-cnvrefs 34901 df-cnvrefrels 34902 |
This theorem is referenced by: elcnvrefrels3 34909 |
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