Theorem tsrss 18542

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 18533	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
2		inss1 4229	. . . . . 6 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
3		dmss 5903	. . . . . 6 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅)
4		ssralv 4051	. . . . . 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 4051	. . . . . . 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 3084	. . . . 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 4230	. . . . . . . . . 10 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
11		dmss 5903	. . . . . . . . . 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 5930	. . . . . . . . 9 ⊢ dom (𝐴 × 𝐴) = 𝐴
14	12, 13	sseqtri 4019	. . . . . . . 8 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴
15	14	sseli 3979	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥 ∈ 𝐴)
16	14	sseli 3979	. . . . . . 7 ⊢ (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦 ∈ 𝐴)
17		brinxp 5755	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
18		brinxp 5755	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
19	18	ancoms 460	. . . . . . . 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 597	. . . . . 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 3176	. . . . 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 3092	. . . 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 614	. 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 2733	. . 3 ⊢ dom 𝑅 = dom 𝑅
27	26	istsr2 18537	. 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
28		eqid 2733	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
29	28	istsr2 18537	. 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 397   ∨ wo 846   ∈ wcel 2107  ∀wral 3062   ∩ cin 3948   ⊆ wss 3949   class class class wbr 5149   × cxp 5675  dom cdm 5677  PosetRelcps 18517   TosetRel ctsr 18518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ps 18519  df-tsr 18520

This theorem is referenced by: (None)