Theorem psssdm2 18639

Description: Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypothesis

Ref	Expression
psssdm.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
psssdm2	⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))

Proof of Theorem psssdm2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmin 5925	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2		psssdm.1	. . . . . 6 ⊢ 𝑋 = dom 𝑅
3	2	eqcomi 2744	. . . . 5 ⊢ dom 𝑅 = 𝑋
4		dmxpid 5944	. . . . 5 ⊢ dom (𝐴 × 𝐴) = 𝐴
5	3, 4	ineq12i 4226	. . . 4 ⊢ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)
6	1, 5	sseqtri 4032	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴)
7	6	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴))
8		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ (𝑋 ∩ 𝐴))
9	8	elin2d 4215	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ 𝐴)
10		elinel1 4211	. . . . 5 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ 𝑋)
11	2	psref 18632	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ 𝑋) → 𝑥𝑅𝑥)
12	10, 11	sylan2 593	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥𝑅𝑥)
13		brinxp2 5766	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
14	9, 9, 12, 13	syl21anbrc 1343	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
15		vex 3482	. . . 4 ⊢ 𝑥 ∈ V
16	15, 15	breldm 5922	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
17	14, 16	syl 17	. 2 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
18	7, 17	eqelssd 4017	1 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∩ cin 3962   ⊆ wss 3963   class class class wbr 5148   × cxp 5687  dom cdm 5689  PosetRelcps 18622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ps 18624

This theorem is referenced by:  psssdm  18640  ordtrest  23226