Theorem sofld 6177

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 5670	. . . . . . . . 9 ⊢ Rel (𝐴 × 𝐴)
2		relss 5759	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
3	1, 2	mpi 21	. . . . . . . 8 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
4	3	ad2antlr 739	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5		df-br 5106	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6		ssun1 4133	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑥})
7		undif1 4433	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
8	6, 7	sseqtrri 3988	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9		simpll 778	. . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ 𝐴)
13	12	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅 ⊆ 𝐴)
14	3	ad2antlr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15		releldm 5925	. . . . . . . . . . . . . . . 16 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
16	14, 15	sylancom 599	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
17	13, 16	sseldd 3940	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐴)
18		sossfld 6176	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
19	9, 17, 18	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20		ssun1 4133	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
21	20, 16	sselid 3937	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
22	21	snssd 4748	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
23	19, 22	unssd 4147	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	8, 23	sstrid 3950	. . . . . . . . . . 11 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
25	24	ex 417	. . . . . . . . . 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 413	. . . . . . . 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 5765	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30		ss0 4359	. . . . . 6 ⊢ (𝑅 ⊆ ∅ → 𝑅 = ∅)
31	29, 30	syl 18	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
32	31	ex 417	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
33	32	necon1ad 2977	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34	33	3impia 1133	. 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 6163	. . . . 5 ⊢ ran (𝐴 × 𝐴) = 𝐴
37	35, 36	sseqtrdi 3979	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ 𝐴)
38	12, 37	unssd 4147	. . 3 ⊢ (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39	38	3ad2ant2 1150	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
40	34, 39	eqssd 3956	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ∖ cdif 3904   ∪ cun 3905   ⊆ wss 3907  ∅c0 4288  {csn 4585  ⟨cop 4591   class class class wbr 5105   Or wor 5559   × cxp 5650  dom cdm 5652  ran crn 5653  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-po 5560  df-so 5561  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by: (None)