Theorem psssdm2 17825

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 5780	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2		psssdm.1	. . . . . 6 ⊢ 𝑋 = dom 𝑅
3	2	eqcomi 2830	. . . . 5 ⊢ dom 𝑅 = 𝑋
4		dmxpid 5800	. . . . 5 ⊢ dom (𝐴 × 𝐴) = 𝐴
5	3, 4	ineq12i 4187	. . . 4 ⊢ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)
6	1, 5	sseqtri 4003	. . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴)
7	6	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴))
8		simpr 487	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ (𝑋 ∩ 𝐴))
9	8	elin2d 4176	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ 𝐴)
10		elinel1 4172	. . . . 5 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ 𝑋)
11	2	psref 17818	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ 𝑋) → 𝑥𝑅𝑥)
12	10, 11	sylan2 594	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥𝑅𝑥)
13		brinxp2 5629	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
14	9, 9, 12, 13	syl21anbrc 1340	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
15		vex 3497	. . . 4 ⊢ 𝑥 ∈ V
16	15, 15	breldm 5777	. . 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 3988	1 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∩ cin 3935   ⊆ wss 3936   class class class wbr 5066   × cxp 5553  dom cdm 5555  PosetRelcps 17808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ps 17810

This theorem is referenced by:  psssdm  17826  ordtrest  21810