Theorem trin2 5438

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 5427	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2		cotr 5427	. . . . . 6 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧))
3		brin 4634	. . . . . . . . . . . . 13 ⊢ (𝑥(𝑅 ∩ 𝑆)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑆𝑦))
4		brin 4634	. . . . . . . . . . . . 13 ⊢ (𝑦(𝑅 ∩ 𝑆)𝑧 ↔ (𝑦𝑅𝑧 ∧ 𝑦𝑆𝑧))
5		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
6		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧))
7	5, 6	anim12d 584	. . . . . . . . . . . . . . 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 865	. . . . . . . . . . . . 13 ⊢ (((𝑥𝑅𝑦 ∧ 𝑥𝑆𝑦) ∧ (𝑦𝑅𝑧 ∧ 𝑦𝑆𝑧)) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
10	3, 4, 9	syl2anb 495	. . . . . . . . . . . 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 4634	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ 𝑆)𝑧 ↔ (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧))
13	11, 12	syl6ibr 241	. . . . . . . . . 10 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
14	13	alanimi 1734	. . . . . . . . 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 1734	. . . . . . . 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 1734	. . . . . . 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 449	. . . . . 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 206	. . . . 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 206	. . 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 444	. 2 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
22		cotr 5427	. 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 223	1 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   ∘ 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-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-co 5047

This theorem is referenced by:  trinxp  5440  trficl  36980