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Theorem joinfval 17267
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 17268 first to reduce net proof size (existence part)?
Hypotheses
Ref Expression
joinfval.u 𝑈 = (lub‘𝐾)
joinfval.j = (join‘𝐾)
Assertion
Ref Expression
joinfval (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
Distinct variable groups:   𝑥,𝑦,𝑧,𝐾   𝑧,𝑈
Allowed substitution hints:   𝑈(𝑥,𝑦)   (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem joinfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3365 . 2 (𝐾𝑉𝐾 ∈ V)
2 joinfval.j . . 3 = (join‘𝐾)
3 fvex 6388 . . . . . . 7 (Base‘𝐾) ∈ V
4 moeq 3535 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
54a1i 11 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))
6 eqid 2765 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}
73, 3, 5, 6oprabex 7354 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} ∈ V
87a1i 11 . . . . 5 (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} ∈ V)
9 joinfval.u . . . . . . . . . . . 12 𝑈 = (lub‘𝐾)
109lubfun 17246 . . . . . . . . . . 11 Fun 𝑈
11 funbrfv2b 6429 . . . . . . . . . . 11 (Fun 𝑈 → ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧)))
1210, 11ax-mp 5 . . . . . . . . . 10 ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧))
13 eqid 2765 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
14 eqid 2765 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘𝐾)
15 simpl 474 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → 𝐾 ∈ V)
16 simpr 477 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ∈ dom 𝑈)
1713, 14, 9, 15, 16lubelss 17248 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
1817ex 401 . . . . . . . . . . . 12 (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19 vex 3353 . . . . . . . . . . . . 13 𝑥 ∈ V
20 vex 3353 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4505 . . . . . . . . . . . 12 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
2218, 21syl6ibr 243 . . . . . . . . . . 11 (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23 eqcom 2772 . . . . . . . . . . . . 13 ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦}))
2423biimpi 207 . . . . . . . . . . . 12 ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦}))
2524a1i 11 . . . . . . . . . . 11 (𝐾 ∈ V → ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦})))
2622, 25anim12d 602 . . . . . . . . . 10 (𝐾 ∈ V → (({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2712, 26syl5bi 233 . . . . . . . . 9 (𝐾 ∈ V → ({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2827alrimiv 2022 . . . . . . . 8 (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2928alrimiv 2022 . . . . . . 7 (𝐾 ∈ V → ∀𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
3029alrimiv 2022 . . . . . 6 (𝐾 ∈ V → ∀𝑥𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
31 ssoprab2 6909 . . . . . 6 (∀𝑥𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
3230, 31syl 17 . . . . 5 (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
338, 32ssexd 4966 . . . 4 (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V)
34 fveq2 6375 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
3534, 9syl6eqr 2817 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
3635breqd 4820 . . . . . 6 (𝑝 = 𝐾 → ({𝑥, 𝑦} (lub‘𝑝)𝑧 ↔ {𝑥, 𝑦}𝑈𝑧))
3736oprabbidv 6907 . . . . 5 (𝑝 = 𝐾 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
38 df-join 17242 . . . . 5 join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
3937, 38fvmptg 6469 . . . 4 ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V) → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
4033, 39mpdan 678 . . 3 (𝐾 ∈ V → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
412, 40syl5eq 2811 . 2 (𝐾 ∈ V → = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
421, 41syl 17 1 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  ∃*wmo 2563  Vcvv 3350  wss 3732  {cpr 4336   class class class wbr 4809  dom cdm 5277  Fun wfun 6062  cfv 6068  {coprab 6843  Basecbs 16130  lecple 16221  lubclub 17208  joincjn 17210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-oprab 6846  df-lub 17240  df-join 17242
This theorem is referenced by:  joinfval2  17268  join0  17404  odumeet  17406  odujoin  17408
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