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Theorem joindm 17224
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 (𝐾𝑉 → 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.
StepHypRef Expression
1 joinfval.u . . . 4 𝑈 = (lub‘𝐾)
2 joinfval.j . . . 4 = (join‘𝐾)
31, 2joinfval2 17223 . . 3 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))})
43dmeqd 5481 . 2 (𝐾𝑉 → dom = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))})
5 dmoprab 6907 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}
6 fvex 6363 . . . . . 6 (𝑈‘{𝑥, 𝑦}) ∈ V
76isseti 3349 . . . . 5 𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
8 19.42v 2030 . . . . 5 (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ ∃𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})))
97, 8mpbiran2 992 . . . 4 (∃𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝑈)
109opabbii 4869 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}
115, 10eqtri 2782 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}
124, 11syl6eq 2810 1 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wex 1853  wcel 2139  {cpr 4323  {copab 4864  dom cdm 5266  cfv 6049  {coprab 6815  lubclub 17163  joincjn 17165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-oprab 6818  df-lub 17195  df-join 17197
This theorem is referenced by:  joindef  17225  joindmss  17228
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