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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 17825 | . . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) | |
2 | cnvps 17816 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel) |
4 | eqid 2821 | . . . . 5 ⊢ dom 𝑅 = dom 𝑅 | |
5 | 4 | istsr 17821 | . . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
6 | 5 | simprbi 499 | . . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)) |
7 | 4 | psrn 17813 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅) |
9 | 8 | sqxpeqd 5582 | . . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅)) |
10 | psrel 17807 | . . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅) |
12 | dfrel2 6041 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
13 | 11, 12 | sylib 220 | . . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅) |
14 | 13 | uneq2d 4139 | . . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅)) |
15 | uncom 4129 | . . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅) | |
16 | 14, 15 | syl6req 2873 | . . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅)) |
17 | 6, 9, 16 | 3sstr3d 4013 | . 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)) |
18 | df-rn 5561 | . . 3 ⊢ ran 𝑅 = dom ◡𝑅 | |
19 | 18 | istsr 17821 | . 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))) |
20 | 3, 17, 19 | sylanbrc 585 | 1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 ⊆ wss 3936 × cxp 5548 ◡ccnv 5549 dom cdm 5550 ran crn 5551 Rel wrel 5555 PosetRelcps 17802 TosetRel ctsr 17803 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ps 17804 df-tsr 17805 |
This theorem is referenced by: ordtbas2 21793 ordtrest2 21806 cnvordtrestixx 31151 |
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