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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsymrel2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.) |
| Ref | Expression |
|---|---|
| dfsymrel2 | ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-symrel 38545 | . 2 ⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | |
| 2 | dfrel6 38348 | . . . . . 6 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) | |
| 3 | 2 | biimpi 216 | . . . . 5 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) |
| 4 | 3 | cnveqd 5886 | . . . 4 ⊢ (Rel 𝑅 → ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ◡𝑅) |
| 5 | 4, 3 | sseq12d 4017 | . . 3 ⊢ (Rel 𝑅 → (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ ◡𝑅 ⊆ 𝑅)) |
| 6 | 5 | pm5.32ri 575 | . 2 ⊢ ((◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) |
| 7 | 1, 6 | bitri 275 | 1 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 Rel wrel 5690 SymRel wsymrel 38194 |
| 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-pr 5432 |
| 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-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-symrel 38545 |
| This theorem is referenced by: dfsymrel3 38551 dfsymrel4 38552 dfsymrel5 38553 elsymrelsrel 38558 symreleq 38559 symrelcoss 38561 refsymrel2 38568 eqvrelsym 38606 refrelredund4 38636 |
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