Theorem sofld 6190

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 5684	. . . . . . . . 9 ⊢ Rel (𝐴 × 𝐴)
2		relss 5773	. . . . . . . . 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 5115	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6		ssun1 4134	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑥})
7		undif1 4440	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
8	6, 7	sseqtrri 3989	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10		dmss 5897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11		dmxpid 5925	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐴 × 𝐴) = 𝐴
12	10, 11	sseqtrdi 3980	. . . . . . . . . . . . . . . 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 5939	. . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
18		sossfld 6189	. . . . . . . . . . . . . 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 4134	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
21	20, 16	sselid 3938	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
22	21	snssd 4757	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
23	19, 22	unssd 4148	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	8, 23	sstrid 3951	. . . . . . . . . . 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 5779	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30		ss0 4362	. . . . . 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 2978	. . 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 5934	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36		rnxpid 6176	. . . . 5 ⊢ ran (𝐴 × 𝐴) = 𝐴
37	35, 36	sseqtrdi 3980	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ 𝐴)
38	12, 37	unssd 4148	. . 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 3957	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 1570   ∈ wcel 2146   ≠ wne 2961   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  ∅c0 4289  {csn 4594  ⟨cop 4600   class class class wbr 5114   Or wor 5573   × cxp 5664  dom cdm 5666  ran crn 5667  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-po 5574  df-so 5575  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677

This theorem is used by: (None)