Theorem psdmrn 18624

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 4131	. . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5964	. . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3946	. . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		pslem 18623	. . . . . 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 3459	. . . . . 6 ⊢ 𝑥 ∈ V
8	7, 7	breldm 5898	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅)
9	6, 8	syl6 36	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅))
10	9	ssrdv 3943	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
11	4, 10	eqssd 3954	. 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
12		ssun2 4132	. . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
13	12, 2	sstri 3946	. . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
14	13	a1i 11	. . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅)
15	7, 7	brelrn 5932	. . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅)
16	6, 15	syl6 36	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅))
17	16	ssrdv 3943	. . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅)
18	14, 17	eqssd 3954	. 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅)
19	11, 18	jca 520	1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∪ cun 3903   ⊆ wss 3905  ∪ cuni 4872   class class class wbr 5109  dom cdm 5661  ran crn 5662  PosetRelcps 18615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ps 18617

This theorem is referenced by:  psref  18625  psrn  18626  psss  18631  tsrdir  18655