Theorem cotr2 14930

Description: Two ways of saying a relation is transitive. Special instance of cotr2g 14929. (Contributed by RP, 22-Mar-2020.)

Hypotheses

Ref	Expression
cotr2.a	⊢ dom 𝑅 ⊆ 𝐴
cotr2.b	⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
cotr2.c	⊢ ran 𝑅 ⊆ 𝐶

Assertion

Ref	Expression
cotr2	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr2

Step	Hyp	Ref	Expression
1		cotr2.a	. 2 ⊢ dom 𝑅 ⊆ 𝐴
2		incom 4138	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅)
3		cotr2.b	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
4	2, 3	eqsstrri 3962	. 2 ⊢ (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵
5		cotr2.c	. 2 ⊢ ran 𝑅 ⊆ 𝐶
6	1, 4, 5	cotr2g 14929	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wral 3053   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5072  dom cdm 5618  ran crn 5619   ∘ ccom 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629

This theorem is referenced by:  cotr3  14931