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Theorem joinfval 17599
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 17600 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 3510 . 2 (𝐾𝑉𝐾 ∈ V)
2 joinfval.j . . 3 = (join‘𝐾)
3 fvex 6676 . . . . . . 7 (Base‘𝐾) ∈ V
4 moeq 3695 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
54a1i 11 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))
6 eqid 2818 . . . . . . 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 7666 . . . . . 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 17578 . . . . . . . . . . 11 Fun 𝑈
11 funbrfv2b 6716 . . . . . . . . . . 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 2818 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
14 eqid 2818 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘𝐾)
15 simpl 483 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → 𝐾 ∈ V)
16 simpr 485 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ∈ dom 𝑈)
1713, 14, 9, 15, 16lubelss 17580 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
1817ex 413 . . . . . . . . . . . 12 (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19 vex 3495 . . . . . . . . . . . . 13 𝑥 ∈ V
20 vex 3495 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4745 . . . . . . . . . . . 12 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
2218, 21syl6ibr 253 . . . . . . . . . . 11 (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23 eqcom 2825 . . . . . . . . . . . 12 ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦}))
2423biimpi 217 . . . . . . . . . . 11 ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦}))
2522, 24anim12d1 609 . . . . . . . . . 10 (𝐾 ∈ V → (({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2612, 25syl5bi 243 . . . . . . . . 9 (𝐾 ∈ V → ({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2726alrimiv 1919 . . . . . . . 8 (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2827alrimiv 1919 . . . . . . 7 (𝐾 ∈ V → ∀𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2928alrimiv 1919 . . . . . 6 (𝐾 ∈ V → ∀𝑥𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
30 ssoprab2 7211 . . . . . 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}))})
3129, 30syl 17 . . . . 5 (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
328, 31ssexd 5219 . . . 4 (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V)
33 fveq2 6663 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
3433, 9syl6eqr 2871 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
3534breqd 5068 . . . . . 6 (𝑝 = 𝐾 → ({𝑥, 𝑦} (lub‘𝑝)𝑧 ↔ {𝑥, 𝑦}𝑈𝑧))
3635oprabbidv 7209 . . . . 5 (𝑝 = 𝐾 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
37 df-join 17574 . . . . 5 join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
3836, 37fvmptg 6759 . . . 4 ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V) → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
3932, 38mpdan 683 . . 3 (𝐾 ∈ V → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
402, 39syl5eq 2865 . 2 (𝐾 ∈ V → = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
411, 40syl 17 1 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  ∃*wmo 2613  Vcvv 3492  wss 3933  {cpr 4559   class class class wbr 5057  dom cdm 5548  Fun wfun 6342  cfv 6348  {coprab 7146  Basecbs 16471  lecple 16560  lubclub 17540  joincjn 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-oprab 7149  df-lub 17572  df-join 17574
This theorem is referenced by:  joinfval2  17600  join0  17736  odumeet  17738  odujoin  17740
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