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 4150	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅)
3		cotr2.b	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
4	2, 3	eqsstrri 3970	. 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 206   ∧ wa 395  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890   class class class wbr 5086  dom cdm 5624  ran crn 5625   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635

This theorem is referenced by:  cotr3  14931