Theorem tsrss 18604

Description: Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)

Assertion

Ref	Expression
tsrss	⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )

Proof of Theorem tsrss

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

Step	Hyp	Ref	Expression
1		psss 18595	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
2		inss1 4217	. . . . . 6 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
3		dmss 5887	. . . . . 6 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅)
4		ssralv 4032	. . . . . 6 ⊢ (dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅 → (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
5	2, 3, 4	mp2b 10	. . . . 5 ⊢ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
6		ssralv 4032	. . . . . . 7 ⊢ (dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅 → (∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
7	2, 3, 6	mp2b 10	. . . . . 6 ⊢ (∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
8	7	ralimi 3074	. . . . 5 ⊢ (∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
9	5, 8	syl 17	. . . 4 ⊢ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
10		inss2 4218	. . . . . . . . . 10 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
11		dmss 5887	. . . . . . . . . 10 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom (𝐴 × 𝐴))
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom (𝐴 × 𝐴)
13		dmxpid 5915	. . . . . . . . 9 ⊢ dom (𝐴 × 𝐴) = 𝐴
14	12, 13	sseqtri 4012	. . . . . . . 8 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴
15	14	sseli 3959	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥 ∈ 𝐴)
16	14	sseli 3959	. . . . . . 7 ⊢ (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦 ∈ 𝐴)
17		brinxp 5738	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
18		brinxp 5738	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
19	18	ancoms 458	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
20	17, 19	orbi12d 918	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
21	15, 16, 20	syl2an 596	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∧ 𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
22	21	ralbidva 3162	. . . . 5 ⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → (∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
23	22	ralbiia 3081	. . . 4 ⊢ (∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
24	9, 23	sylib 218	. . 3 ⊢ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
25	1, 24	anim12i 613	. 2 ⊢ ((𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ∧ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
26		eqid 2736	. . 3 ⊢ dom 𝑅 = dom 𝑅
27	26	istsr2 18599	. 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
28		eqid 2736	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
29	28	istsr2 18599	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel ↔ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ∧ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
30	25, 27, 29	3imtr4i 292	1 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∈ wcel 2109  ∀wral 3052   ∩ cin 3930   ⊆ wss 3931   class class class wbr 5124   × cxp 5657  dom cdm 5659  PosetRelcps 18579   TosetRel ctsr 18580

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-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ps 18581  df-tsr 18582

This theorem is referenced by: (None)