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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 14336. (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 4178 | . . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅) | |
3 | cotr2.b | . . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 | |
4 | 2, 3 | eqsstrri 4002 | . 2 ⊢ (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵 |
5 | cotr2.c | . 2 ⊢ ran 𝑅 ⊆ 𝐶 | |
6 | 1, 4, 5 | cotr2g 14336 | 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 ∀wral 3138 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ran crn 5556 ∘ ccom 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 |
This theorem is referenced by: cotr3 14338 |
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