Theorem joindm 18008

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

Hypotheses

Ref	Expression
joinfval.u	⊢ 𝑈 = (lub‘𝐾)
joinfval.j	⊢ ∨ = (join‘𝐾)

Assertion

Ref	Expression
joindm	⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})

Distinct variable group:   𝑥,𝑦,𝐾

Allowed substitution hints:   𝑈(𝑥,𝑦)   ∨ (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem joindm

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joinfval.u	. . . 4 ⊢ 𝑈 = (lub‘𝐾)
2		joinfval.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3	1, 2	joinfval2 18007	. . 3 ⊢ (𝐾 ∈ 𝑉 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
4	3	dmeqd 5803	. 2 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
5		dmoprab 7354	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}
6		fvex 6769	. . . . . 6 ⊢ (𝑈‘{𝑥, 𝑦}) ∈ V
7	6	isseti 3437	. . . . 5 ⊢ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
8		19.42v 1958	. . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})))
9	7, 8	mpbiran2 706	. . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝑈)
10	9	opabbii 5137	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}
11	5, 10	eqtri 2766	. 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}
12	4, 11	eqtrdi 2795	1 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cpr 4560  {copab 5132  dom cdm 5580  ‘cfv 6418  {coprab 7256  lubclub 17942  joincjn 17944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-oprab 7259  df-lub 17979  df-join 17981

This theorem is referenced by:  joindef  18009  joindmss  18012