Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsymrel3 | Structured version Visualization version GIF version |
Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.) |
Ref | Expression |
---|---|
dfsymrel3 | ⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsymrel2 36689 | . 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
2 | cnvsym 6022 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
3 | 2 | anbi1i 623 | . 2 ⊢ ((◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ Rel 𝑅)) |
4 | 1, 3 | bitri 274 | 1 ⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1535 ⊆ wss 3889 class class class wbr 5077 ◡ccnv 5590 Rel wrel 5596 SymRel wsymrel 36373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-br 5078 df-opab 5140 df-xp 5597 df-rel 5598 df-cnv 5599 df-dm 5601 df-rn 5602 df-res 5603 df-symrel 36684 |
This theorem is referenced by: refsymrel3 36708 |
Copyright terms: Public domain | W3C validator |