Theorem sofld 6169

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 5663	. . . . . . . . 9 ⊢ Rel (𝐴 × 𝐴)
2		relss 5752	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
3	1, 2	mpi 20	. . . . . . . 8 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
4	3	ad2antlr 737	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5		df-br 5100	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6		ssun1 4130	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑥})
7		undif1 4429	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
8	6, 7	sseqtrri 3985	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9		simpll 776	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10		dmss 5876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11		dmxpid 5904	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐴 × 𝐴) = 𝐴
12	10, 11	sseqtrdi 3976	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ 𝐴)
13	12	ad2antlr 737	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅 ⊆ 𝐴)
14	3	ad2antlr 737	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15		releldm 5918	. . . . . . . . . . . . . . . 16 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
16	14, 15	sylancom 597	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
17	13, 16	sseldd 3937	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
18		sossfld 6168	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
19	9, 17, 18	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20		ssun1 4130	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
21	20, 16	sselid 3934	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
22	21	snssd 4744	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
23	19, 22	unssd 4144	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	8, 23	sstrid 3947	. . . . . . . . . . 11 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
25	24	ex 416	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦 → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
26	5, 25	biimtrrid 245	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
27	26	con3dimp 412	. . . . . . . 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 5758	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30		ss0 4355	. . . . . 6 ⊢ (𝑅 ⊆ ∅ → 𝑅 = ∅)
31	29, 30	syl 17	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
32	31	ex 416	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
33	32	necon1ad 2973	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34	33	3impia 1129	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35		rnss 5913	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36		rnxpid 6155	. . . . 5 ⊢ ran (𝐴 × 𝐴) = 𝐴
37	35, 36	sseqtrdi 3976	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ 𝐴)
38	12, 37	unssd 4144	. . 3 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39	38	3ad2ant2 1146	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
40	34, 39	eqssd 3953	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4581  ⟨cop 4587   class class class wbr 5099   Or wor 5552   × cxp 5643  dom cdm 5645  ran crn 5646  Rel wrel 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-po 5553  df-so 5554  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656

This theorem is referenced by: (None)