Theorem trin2 5096

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 5086	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2		cotr 5086	. . . . . 6 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧))
3		brin 4115	. . . . . . . . . . . . 13 ⊢ (𝑥(𝑅 ∩ 𝑆)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑆𝑦))
4		brin 4115	. . . . . . . . . . . . 13 ⊢ (𝑦(𝑅 ∩ 𝑆)𝑧 ↔ (𝑦𝑅𝑧 ∧ 𝑦𝑆𝑧))
5		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
6		simpl 109	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧))
7	5, 6	anim12d 335	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
8	7	com12 30	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧)) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
9	8	an4s 590	. . . . . . . . . . . . 13 ⊢ (((𝑥𝑅𝑦 ∧ 𝑥𝑆𝑦) ∧ (𝑦𝑅𝑧 ∧ 𝑦𝑆𝑧)) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
10	3, 4, 9	syl2anb 291	. . . . . . . . . . . 12 ⊢ ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
11	10	com12 30	. . . . . . . . . . 11 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧)))
12		brin 4115	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ 𝑆)𝑧 ↔ (𝑥𝑅𝑧 ∧ 𝑥𝑆𝑧))
13	11, 12	imbitrrdi 162	. . . . . . . . . 10 ⊢ ((((𝑥𝑆𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑆𝑧) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
14	13	alanimi 1485	. . . . . . . . 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 1485	. . . . . . . 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 1485	. . . . . . 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 115	. . . . . 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 121	. . . . 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 30	. . . 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 121	. . 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 124	. 2 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ∩ 𝑆)𝑦 ∧ 𝑦(𝑅 ∩ 𝑆)𝑧) → 𝑥(𝑅 ∩ 𝑆)𝑧))
22		cotr 5086	. 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 134	1 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1373   ∩ cin 3176   ⊆ wss 3177   class class class wbr 4062   ∘ ccom 4700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-rel 4703  df-co 4705

This theorem is referenced by:  trinxp  5098