Theorem psssdm 18534

Description: Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)

Hypothesis

Ref	Expression
psssdm.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
psssdm	⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)

Proof of Theorem psssdm

Step	Hyp	Ref	Expression
1		psssdm.1	. . 3 ⊢ 𝑋 = dom 𝑅
2	1	psssdm2 18533	. 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))
3		sseqin2 4215	. . 3 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
4	3	biimpi 215	. 2 ⊢ (𝐴 ⊆ 𝑋 → (𝑋 ∩ 𝐴) = 𝐴)
5	2, 4	sylan9eq 2792	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   ⊆ wss 3948   × cxp 5674  dom cdm 5676  PosetRelcps 18516

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ps 18518

This theorem is referenced by:  ordtrest2lem  22706  ordtrest2  22707  icopnfhmeo  24458  iccpnfhmeo  24460  xrhmeo  24461  xrge0iifhmeo  32911