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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 3660 | . 2 ⊢ dom 𝑅 ⊆ 𝐴 |
3 | cotr3.b | . . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶) | |
4 | cotr3.c | . . . . 5 ⊢ 𝐶 = ran 𝑅 | |
5 | 1, 4 | ineq12i 3812 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅) |
6 | 3, 5 | eqtri 2644 | . . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅) |
7 | 6 | eqimss2i 3660 | . 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 |
8 | 4 | eqimss2i 3660 | . 2 ⊢ ran 𝑅 ⊆ 𝐶 |
9 | 2, 7, 8 | cotr2 13716 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∀wral 2912 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ran crn 5115 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 |
This theorem is referenced by: (None) |
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