Theorem sofld 6218

Description: The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
sofld	⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

Proof of Theorem sofld

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

Step	Hyp	Ref	Expression
1		relxp 5718	. . . . . . . . 9 ⊢ Rel (𝐴 × 𝐴)
2		relss 5805	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
3	1, 2	mpi 20	. . . . . . . 8 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
4	3	ad2antlr 726	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5		df-br 5167	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6		ssun1 4201	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑥})
7		undif1 4499	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
8	6, 7	sseqtrri 4046	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10		dmss 5927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11		dmxpid 5955	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐴 × 𝐴) = 𝐴
12	10, 11	sseqtrdi 4059	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ 𝐴)
13	12	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅 ⊆ 𝐴)
14	3	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15		releldm 5969	. . . . . . . . . . . . . . . 16 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
16	14, 15	sylancom 587	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
17	13, 16	sseldd 4009	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
18		sossfld 6217	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
19	9, 17, 18	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20		ssun1 4201	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
21	20, 16	sselid 4006	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
22	21	snssd 4834	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
23	19, 22	unssd 4215	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	8, 23	sstrid 4020	. . . . . . . . . . 11 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
25	24	ex 412	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦 → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
26	5, 25	biimtrrid 243	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
27	26	con3dimp 408	. . . . . . . 8 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
28	27	pm2.21d 121	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ ∅))
29	4, 28	relssdv 5812	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30		ss0 4425	. . . . . 6 ⊢ (𝑅 ⊆ ∅ → 𝑅 = ∅)
31	29, 30	syl 17	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
32	31	ex 412	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
33	32	necon1ad 2963	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34	33	3impia 1117	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35		rnss 5964	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36		rnxpid 6204	. . . . 5 ⊢ ran (𝐴 × 𝐴) = 𝐴
37	35, 36	sseqtrdi 4059	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ 𝐴)
38	12, 37	unssd 4215	. . 3 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39	38	3ad2ant2 1134	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
40	34, 39	eqssd 4026	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  {csn 4648  ⟨cop 4654   class class class wbr 5166   Or wor 5606   × cxp 5698  dom cdm 5700  ran crn 5701  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by: (None)