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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 4025 | . 2 ⊢ dom 𝑅 ⊆ 𝐴 |
3 | cotr3.b | . . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶) | |
4 | cotr3.c | . . . . 5 ⊢ 𝐶 = ran 𝑅 | |
5 | 1, 4 | ineq12i 4186 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅) |
6 | 3, 5 | eqtri 2844 | . . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅) |
7 | 6 | eqimss2i 4025 | . 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 |
8 | 4 | eqimss2i 4025 | . 2 ⊢ ran 𝑅 ⊆ 𝐶 |
9 | 2, 7, 8 | cotr2 14331 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∀wral 3138 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 dom cdm 5549 ran crn 5550 ∘ ccom 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 |
This theorem is referenced by: (None) |
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