Theorem cotr2 14949

Description: Two ways of saying a relation is transitive. Special instance of cotr2g 14948. (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 4174	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) = (ran 𝑅 ∩ dom 𝑅)
3		cotr2.b	. . 3 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
4	2, 3	eqsstrri 3996	. 2 ⊢ (ran 𝑅 ∩ dom 𝑅) ⊆ 𝐵
5		cotr2.c	. 2 ⊢ ran 𝑅 ⊆ 𝐶
6	1, 4, 5	cotr2g 14948	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wral 3045   ∩ cin 3915   ⊆ wss 3916   class class class wbr 5109  dom cdm 5640  ran crn 5641   ∘ ccom 5644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651

This theorem is referenced by:  cotr3  14950