Theorem tsrss 18572

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 18563	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
2		inss1 4224	. . . . . 6 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
3		dmss 5899	. . . . . 6 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅)
4		ssralv 4046	. . . . . 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 4046	. . . . . . 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 3078	. . . . 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 4225	. . . . . . . . . 10 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
11		dmss 5899	. . . . . . . . . 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 5926	. . . . . . . . 9 ⊢ dom (𝐴 × 𝐴) = 𝐴
14	12, 13	sseqtri 4014	. . . . . . . 8 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴
15	14	sseli 3974	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥 ∈ 𝐴)
16	14	sseli 3974	. . . . . . 7 ⊢ (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦 ∈ 𝐴)
17		brinxp 5750	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
18		brinxp 5750	. . . . . . . . 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 917	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
21	15, 16, 20	syl2an 595	. . . . . 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 3170	. . . . 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 3086	. . . 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 217	. . 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 612	. 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 2727	. . 3 ⊢ dom 𝑅 = dom 𝑅
27	26	istsr2 18567	. 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
28		eqid 2727	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
29	28	istsr2 18567	. 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 205   ∧ wa 395   ∨ wo 846   ∈ wcel 2099  ∀wral 3056   ∩ cin 3943   ⊆ wss 3944   class class class wbr 5142   × cxp 5670  dom cdm 5672  PosetRelcps 18547   TosetRel ctsr 18548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ps 18549  df-tsr 18550

This theorem is referenced by: (None)