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Mirrors > Home > MPE Home > Th. List > cnvtsr | Structured version Visualization version GIF version |
Description: The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
cnvtsr | ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsrps 18645 | . . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) | |
2 | cnvps 18636 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel) |
4 | eqid 2735 | . . . . 5 ⊢ dom 𝑅 = dom 𝑅 | |
5 | 4 | istsr 18641 | . . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
6 | 5 | simprbi 496 | . . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)) |
7 | 4 | psrn 18633 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅) |
9 | 8 | sqxpeqd 5721 | . . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅)) |
10 | psrel 18627 | . . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅) |
12 | dfrel2 6211 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
13 | 11, 12 | sylib 218 | . . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅) |
14 | 13 | uneq2d 4178 | . . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅)) |
15 | uncom 4168 | . . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅) | |
16 | 14, 15 | eqtr2di 2792 | . . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅)) |
17 | 6, 9, 16 | 3sstr3d 4042 | . 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)) |
18 | df-rn 5700 | . . 3 ⊢ ran 𝑅 = dom ◡𝑅 | |
19 | 18 | istsr 18641 | . 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))) |
20 | 3, 17, 19 | sylanbrc 583 | 1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 × cxp 5687 ◡ccnv 5688 dom cdm 5689 ran crn 5690 Rel wrel 5694 PosetRelcps 18622 TosetRel ctsr 18623 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ps 18624 df-tsr 18625 |
This theorem is referenced by: ordtbas2 23215 ordtrest2 23228 cnvordtrestixx 33874 |
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