Theorem joindm 18297

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 18296	. . 3 ⊢ (𝐾 ∈ 𝑉 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
4	3	dmeqd 5852	. 2 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
5		dmoprab 7456	. . 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 6839	. . . . . 6 ⊢ (𝑈‘{𝑥, 𝑦}) ∈ V
7	6	isseti 3456	. . . . 5 ⊢ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
8		19.42v 1953	. . . . 5 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})))
9	7, 8	mpbiran2 710	. . . 4 ⊢ (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝑈)
10	9	opabbii 5162	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}
11	5, 10	eqtri 2752	. 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}
12	4, 11	eqtrdi 2780	1 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cpr 4581  {copab 5157  dom cdm 5623  ‘cfv 6486  {coprab 7354  lubclub 18233  joincjn 18235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-oprab 7357  df-lub 18268  df-join 18270

This theorem is referenced by:  joindef  18298  joindmss  18301