Theorem cotr2 13564

Description: Two ways of saying a relation is transitive. Special instance of cotr2g 13563. (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 3767	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅)
3		cotr2.b	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
4	2, 3	eqsstr3i 3599	. 2 ⊢ (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵
5		cotr2.c	. 2 ⊢ ran 𝑅 ⊆ 𝐶
6	1, 4, 5	cotr2g 13563	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  ran crn 5039   ∘ ccom 5042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049

This theorem is referenced by:  cotr3  13565