Theorem psdmrn 18530

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 4107	. . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5916	. . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3924	. . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		pslem 18529	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥)))
6	5	simp2d 1149	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥))
7		vex 3435	. . . . . 6 ⊢ 𝑥 ∈ V
8	7, 7	breldm 5850	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
9	6, 8	syl6 35	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅))
10	9	ssrdv 3921	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
11	4, 10	eqssd 3932	. 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
12		ssun2 4108	. . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
13	12, 2	sstri 3924	. . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
14	13	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅)
15	7, 7	brelrn 5884	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅)
16	6, 15	syl6 35	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅))
17	16	ssrdv 3921	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅)
18	14, 17	eqssd 3932	. 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅)
19	11, 18	jca 516	1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∪ cun 3881   ⊆ wss 3883  ∪ cuni 4838   class class class wbr 5072  dom cdm 5618  ran crn 5619  PosetRelcps 18521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ps 18523

This theorem is referenced by:  psref  18531  psrn  18532  psss  18537  tsrdir  18561