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Theorem joinfval 16929
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 16930 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 3201 . 2 (𝐾𝑉𝐾 ∈ V)
2 joinfval.j . . 3 = (join‘𝐾)
3 fvex 6163 . . . . . . 7 (Base‘𝐾) ∈ V
4 moeq 3368 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
54a1i 11 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))
6 eqid 2621 . . . . . . 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 7108 . . . . . 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 16908 . . . . . . . . . . 11 Fun 𝑈
11 funbrfv2b 6202 . . . . . . . . . . 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 2621 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
14 eqid 2621 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘𝐾)
15 simpl 473 . . . . . . . . . . . . . 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 16910 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
1817ex 450 . . . . . . . . . . . 12 (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19 vex 3192 . . . . . . . . . . . . 13 𝑥 ∈ V
20 vex 3192 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4324 . . . . . . . . . . . 12 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
2218, 21syl6ibr 242 . . . . . . . . . . 11 (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23 eqcom 2628 . . . . . . . . . . . . 13 ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦}))
2423biimpi 206 . . . . . . . . . . . 12 ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦}))
2524a1i 11 . . . . . . . . . . 11 (𝐾 ∈ V → ((𝑈‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝑈‘{𝑥, 𝑦})))
2622, 25anim12d 585 . . . . . . . . . 10 (𝐾 ∈ V → (({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2712, 26syl5bi 232 . . . . . . . . 9 (𝐾 ∈ V → ({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2827alrimiv 1852 . . . . . . . 8 (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
2928alrimiv 1852 . . . . . . 7 (𝐾 ∈ V → ∀𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
3029alrimiv 1852 . . . . . 6 (𝐾 ∈ V → ∀𝑥𝑦𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
31 ssoprab2 6671 . . . . . 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 4770 . . . 4 (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V)
34 fveq2 6153 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
3534, 9syl6eqr 2673 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
3635breqd 4629 . . . . . 6 (𝑝 = 𝐾 → ({𝑥, 𝑦} (lub‘𝑝)𝑧 ↔ {𝑥, 𝑦}𝑈𝑧))
3736oprabbidv 6669 . . . . 5 (𝑝 = 𝐾 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
38 df-join 16904 . . . . 5 join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
3937, 38fvmptg 6242 . . . 4 ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V) → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
4033, 39mpdan 701 . . 3 (𝐾 ∈ V → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
412, 40syl5eq 2667 . 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 196  wa 384  wal 1478   = wceq 1480  wcel 1987  ∃*wmo 2470  Vcvv 3189  wss 3559  {cpr 4155   class class class wbr 4618  dom cdm 5079  Fun wfun 5846  cfv 5852  {coprab 6611  Basecbs 15788  lecple 15876  lubclub 16870  joincjn 16872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-oprab 6614  df-lub 16902  df-join 16904
This theorem is referenced by:  joinfval2  16930  join0  17066  odumeet  17068  odujoin  17070
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