Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cotr2 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is transitive. Special instance of cotr2g 14544. (Contributed by RP, 22-Mar-2020.) |
Ref | Expression |
---|---|
cotr2.a | ⊢ dom 𝑅 ⊆ 𝐴 |
cotr2.b | ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 |
cotr2.c | ⊢ ran 𝑅 ⊆ 𝐶 |
Ref | Expression |
---|---|
cotr2 | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cotr2.a | . 2 ⊢ dom 𝑅 ⊆ 𝐴 | |
2 | incom 4120 | . . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅) | |
3 | cotr2.b | . . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 | |
4 | 2, 3 | eqsstrri 3941 | . 2 ⊢ (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵 |
5 | cotr2.c | . 2 ⊢ ran 𝑅 ⊆ 𝐶 | |
6 | 1, 4, 5 | cotr2g 14544 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wral 3061 ∩ cin 3870 ⊆ wss 3871 class class class wbr 5058 dom cdm 5556 ran crn 5557 ∘ ccom 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rab 3070 df-v 3415 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-sn 4547 df-pr 4549 df-op 4553 df-br 5059 df-opab 5121 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 |
This theorem is referenced by: cotr3 14546 |
Copyright terms: Public domain | W3C validator |