Theorem psdmrn 18664

Description: The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)

Assertion

Ref	Expression
psdmrn	⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))

Proof of Theorem psdmrn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun1 4124	. . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5958	. . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3940	. . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		pslem 18663	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥)))
6	5	simp2d 1161	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥))
7		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
8	7, 7	breldm 5892	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
9	6, 8	syl6 36	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅))
10	9	ssrdv 3937	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
11	4, 10	eqssd 3948	. 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
12		ssun2 4125	. . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
13	12, 2	sstri 3940	. . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
14	13	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅)
15	7, 7	brelrn 5926	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅)
16	6, 15	syl6 36	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅))
17	16	ssrdv 3937	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅)
18	14, 17	eqssd 3948	. 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅)
19	11, 18	jca 521	1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3897   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  dom cdm 5655  ran crn 5656  PosetRelcps 18655

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ps 18657

This theorem is used by:  psref  18665  psrn  18666  psss  18671  tsrdir  18695