Theorem psssdm2 18513

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 5900	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2		psssdm.1	. . . . . 6 ⊢ 𝑋 = dom 𝑅
3	2	eqcomi 2740	. . . . 5 ⊢ dom 𝑅 = 𝑋
4		dmxpid 5918	. . . . 5 ⊢ dom (𝐴 × 𝐴) = 𝐴
5	3, 4	ineq12i 4203	. . . 4 ⊢ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)
6	1, 5	sseqtri 4011	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴)
7	6	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴))
8		simpr 485	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ (𝑋 ∩ 𝐴))
9	8	elin2d 4192	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ 𝐴)
10		elinel1 4188	. . . . 5 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ 𝑋)
11	2	psref 18506	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ 𝑋) → 𝑥𝑅𝑥)
12	10, 11	sylan2 593	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥𝑅𝑥)
13		brinxp2 5742	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
14	9, 9, 12, 13	syl21anbrc 1344	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
15		vex 3474	. . . 4 ⊢ 𝑥 ∈ V
16	15, 15	breldm 5897	. . 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 3996	1 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3940   ⊆ wss 3941   class class class wbr 5138   × cxp 5664  dom cdm 5666  PosetRelcps 18496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ps 18498

This theorem is referenced by:  psssdm  18514  ordtrest  22630