Theorem sofld 6178

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 5668	. . . . . . . . 9 ⊢ Rel (𝐴 × 𝐴)
2		relss 5757	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
3	1, 2	mpi 21	. . . . . . . 8 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
4	3	ad2antlr 740	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5		df-br 5103	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6		ssun1 4176	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑥})
7		undif1 4429	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
8	6, 7	sseqtrri 3957	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10		dmss 5883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11		dmxpid 5911	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐴 × 𝐴) = 𝐴
12	10, 11	sseqtrdi 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ 𝐴)
13	12	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅 ⊆ 𝐴)
14	3	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15		releldm 5925	. . . . . . . . . . . . . . . 16 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
16	14, 15	sylancom 600	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
17	13, 16	sseldd 3909	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
18		sossfld 6177	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
19	9, 17, 18	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20		ssun1 4176	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
21	20, 16	sselid 3906	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
22	21	snssd 4743	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
23	19, 22	unssd 4190	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	8, 23	sstrid 3919	. . . . . . . . . . 11 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
25	24	ex 418	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦 → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
26	5, 25	biimtrrid 246	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
27	26	con3dimp 414	. . . . . . . 8 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
28	27	pm2.21d 122	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ ∅))
29	4, 28	relssdv 5763	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30		ss0 4351	. . . . . 6 ⊢ (𝑅 ⊆ ∅ → 𝑅 = ∅)
31	29, 30	syl 18	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
32	31	ex 418	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
33	32	necon1ad 2972	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34	33	3impia 1135	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35		rnss 5920	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36		rnxpid 6164	. . . . 5 ⊢ ran (𝐴 × 𝐴) = 𝐴
37	35, 36	sseqtrdi 3948	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ 𝐴)
38	12, 37	unssd 4190	. . 3 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39	38	3ad2ant2 1152	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
40	34, 39	eqssd 3925	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  {csn 4584  ⟨cop 4834   class class class wbr 5102   Or wor 5557   × cxp 5648  dom cdm 5650  ran crn 5651  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by: (None)