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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 35768. (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 35779 | . 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | |
2 | df-cnvrefs 35778 | . 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | dmexg 7613 | . . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V) | |
4 | 3 | elv 3499 | . . . . 5 ⊢ dom 𝑟 ∈ V |
5 | rnexg 7614 | . . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V) | |
6 | 5 | elv 3499 | . . . . 5 ⊢ ran 𝑟 ∈ V |
7 | 4, 6 | xpex 7476 | . . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V |
8 | inex2g 5224 | . . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
9 | brcnvssr 35761 | . . . 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 35745 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
12 | 11 | elv 3499 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
13 | 12 | biimpi 218 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
14 | 13 | sseq1d 3998 | . . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))) |
15 | 10, 14 | syl5bb 285 | . 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 35530 | 1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 I cid 5459 × cxp 5553 ◡ccnv 5554 dom cdm 5555 ran crn 5556 Rels crels 35470 S cssr 35471 CnvRefs ccnvrefs 35475 CnvRefRels ccnvrefrels 35476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-rels 35740 df-ssr 35753 df-cnvrefs 35778 df-cnvrefrels 35779 |
This theorem is referenced by: elcnvrefrels2 35785 |
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