Theorem trin2 6146

Description: The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)

Assertion

Ref	Expression
trin2	⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))

Proof of Theorem trin2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cotr 6133	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2		cotr 6133	. . . . . 6 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧))
3		brin 5200	. . . . . . . . . . . . 13 ⊢ (𝑥(𝑅 ∩ 𝑆)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑆𝑦))
4		brin 5200	. . . . . . . . . . . . 13 ⊢ (𝑦(𝑅 ∩ 𝑆)𝑧 ↔ (𝑦𝑅𝑧 ∧ 𝑦𝑆𝑧))
5		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
6		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧))
7	5, 6	anim12d 609	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
8	7	com12 32	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧)) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
9	8	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑥𝑅𝑦 ∧ 𝑥𝑆𝑦) ∧ (𝑦𝑅𝑧 ∧ 𝑦𝑆𝑧)) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
10	3, 4, 9	syl2anb 598	. . . . . . . . . . . 12 ⊢ ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
11	10	com12 32	. . . . . . . . . . 11 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
12		brin 5200	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ 𝑆)𝑧 ↔ (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧))
13	11, 12	imbitrrdi 252	. . . . . . . . . 10 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
14	13	alanimi 1813	. . . . . . . . 9 ⊢ ((∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
15	14	alanimi 1813	. . . . . . . 8 ⊢ ((∀𝑦∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
16	15	alanimi 1813	. . . . . . 7 ⊢ ((∀𝑥∀𝑦∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
17	16	ex 412	. . . . . 6 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧)))
18	2, 17	sylbi 217	. . . . 5 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧)))
19	18	com12 32	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧)))
20	1, 19	sylbi 217	. . 3 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧)))
21	20	imp 406	. 2 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
22		cotr 6133	. 2 ⊢ (((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆) ↔ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
23	21, 22	sylibr 234	1 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   ∩ cin 3962   ⊆ wss 3963   class class class wbr 5148   ∘ ccom 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-co 5698

This theorem is referenced by:  trinxp  6148  trficl  43659