Theorem tsrss 17223

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 17214	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
2		inss1 3833	. . . . . 6 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
3		dmss 5323	. . . . . 6 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅)
4		ssralv 3666	. . . . . 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 3666	. . . . . . 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 2952	. . . . 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 3834	. . . . . . . . . 10 ⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
11		dmss 5323	. . . . . . . . . 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 5345	. . . . . . . . 9 ⊢ dom (𝐴 × 𝐴) = 𝐴
14	12, 13	sseqtri 3637	. . . . . . . 8 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴
15	14	sseli 3599	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥 ∈ 𝐴)
16	14	sseli 3599	. . . . . . 7 ⊢ (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦 ∈ 𝐴)
17		brinxp 5181	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
18		brinxp 5181	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
19	18	ancoms 469	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
20	17, 19	orbi12d 746	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
21	15, 16, 20	syl2an 494	. . . . . 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 2985	. . . . 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 2979	. . . 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 208	. . 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 590	. 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 2622	. . 3 ⊢ dom 𝑅 = dom 𝑅
27	26	istsr2 17218	. 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
28		eqid 2622	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
29	28	istsr2 17218	. 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 281	1 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653   × cxp 5112  dom cdm 5114  PosetRelcps 17198   TosetRel ctsr 17199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ps 17200  df-tsr 17201

This theorem is referenced by: (None)