Theorem psdmrn 18581

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 4153	. . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5953	. . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3968	. . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		pslem 18580	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥)))
6	5	simp2d 1143	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥))
7		vex 3463	. . . . . 6 ⊢ 𝑥 ∈ V
8	7, 7	breldm 5888	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
9	6, 8	syl6 35	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅))
10	9	ssrdv 3964	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
11	4, 10	eqssd 3976	. 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
12		ssun2 4154	. . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
13	12, 2	sstri 3968	. . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
14	13	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅)
15	7, 7	brelrn 5922	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅)
16	6, 15	syl6 35	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅))
17	16	ssrdv 3964	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅)
18	14, 17	eqssd 3976	. 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅)
19	11, 18	jca 511	1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∪ cun 3924   ⊆ wss 3926  ∪ cuni 4883   class class class wbr 5119  dom cdm 5654  ran crn 5655  PosetRelcps 18572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ps 18574

This theorem is referenced by:  psref  18582  psrn  18583  psss  18588  tsrdir  18612