Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cotr3 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.) |
Ref | Expression |
---|---|
cotr3.a | ⊢ 𝐴 = dom 𝑅 |
cotr3.b | ⊢ 𝐵 = (𝐴 ∩ 𝐶) |
cotr3.c | ⊢ 𝐶 = ran 𝑅 |
Ref | Expression |
---|---|
cotr3 | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cotr3.a | . . 3 ⊢ 𝐴 = dom 𝑅 | |
2 | 1 | eqimss2i 3623 | . 2 ⊢ dom 𝑅 ⊆ 𝐴 |
3 | cotr3.b | . . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶) | |
4 | cotr3.c | . . . . 5 ⊢ 𝐶 = ran 𝑅 | |
5 | 1, 4 | ineq12i 3774 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅) |
6 | 3, 5 | eqtri 2632 | . . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅) |
7 | 6 | eqimss2i 3623 | . 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 |
8 | 4 | eqimss2i 3623 | . 2 ⊢ ran 𝑅 ⊆ 𝐶 |
9 | 2, 7, 8 | cotr2 13564 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∀wral 2896 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 ran crn 5039 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |