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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 18632 | . . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) | |
| 2 | cnvps 18623 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel) |
| 4 | eqid 2737 | . . . . 5 ⊢ dom 𝑅 = dom 𝑅 | |
| 5 | 4 | istsr 18628 | . . . 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 18620 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅) |
| 8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅) |
| 9 | 8 | sqxpeqd 5717 | . . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅)) |
| 10 | psrel 18614 | . . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
| 11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅) |
| 12 | dfrel2 6209 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
| 13 | 11, 12 | sylib 218 | . . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅) |
| 14 | 13 | uneq2d 4168 | . . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅)) |
| 15 | uncom 4158 | . . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅) | |
| 16 | 14, 15 | eqtr2di 2794 | . . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅)) |
| 17 | 6, 9, 16 | 3sstr3d 4038 | . 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)) |
| 18 | df-rn 5696 | . . 3 ⊢ ran 𝑅 = dom ◡𝑅 | |
| 19 | 18 | istsr 18628 | . 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 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 Rel wrel 5690 PosetRelcps 18609 TosetRel ctsr 18610 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ps 18611 df-tsr 18612 |
| This theorem is referenced by: ordtbas2 23199 ordtrest2 23212 cnvordtrestixx 33912 |
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