Theorem meetdm 18414

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 18413	. . 3 ⊢ (𝐾 ∈ 𝑉 → ∧ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
4	3	dmeqd 5912	. 2 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))})
5		dmoprab 7527	. . 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 6914	. . . . . 6 ⊢ (𝐺‘{𝑥, 𝑦}) ∈ V
7	6	isseti 3479	. . . . 5 ⊢ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
8		19.42v 1950	. . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})))
9	7, 8	mpbiran2 708	. . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝐺)
10	9	opabbii 5220	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
11	5, 10	eqtri 2754	. 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
12	4, 11	eqtrdi 2782	1 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cpr 4635  {copab 5215  dom cdm 5682  ‘cfv 6554  {coprab 7425  glbcglb 18335  meetcmee 18337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-oprab 7428  df-glb 18372  df-meet 18374

This theorem is referenced by:  meetdef  18415  meetdmss  18418