Theorem meetdm 18308

Description: Domain of meet function for a poset-type structure. (Contributed by NM, 16-Sep-2018.)

Hypotheses

Ref	Expression
meetfval.u	⊢ 𝐺 = (glb‘𝐾)
meetfval.m	⊢ ∧ = (meet‘𝐾)

Assertion

Ref	Expression
meetdm	⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})

Distinct variable group:   𝑥,𝑦,𝐾

Allowed substitution hints:   𝐺(𝑥,𝑦)   ∧ (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem meetdm

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meetfval.u	. . . 4 ⊢ 𝐺 = (glb‘𝐾)
2		meetfval.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
3	1, 2	meetfval2 18307	. . 3 ⊢ (𝐾 ∈ 𝑉 → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
4	3	dmeqd 5852	. 2 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
5		dmoprab 7459	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}
6		fvex 6845	. . . . . 6 ⊢ (𝐺‘{𝑥, 𝑦}) ∈ V
7	6	isseti 3456	. . . . 5 ⊢ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
8		19.42v 1954	. . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})))
9	7, 8	mpbiran2 710	. . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝐺)
10	9	opabbii 5163	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
11	5, 10	eqtri 2757	. 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
12	4, 11	eqtrdi 2785	1 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cpr 4580  {copab 5158  dom cdm 5622  ‘cfv 6490  {coprab 7357  glbcglb 18231  meetcmee 18233

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-oprab 7360  df-glb 18266  df-meet 18268

This theorem is referenced by:  meetdef  18309  meetdmss  18312