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Theorem joinfval 18328
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 18329 first to reduce net proof size (existence part)?
Hypotheses
Ref Expression
joinfval.u π‘ˆ = (lubβ€˜πΎ)
joinfval.j ∨ = (joinβ€˜πΎ)
Assertion
Ref Expression
joinfval (𝐾 ∈ 𝑉 β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐾   𝑧,π‘ˆ
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   ∨ (π‘₯,𝑦,𝑧)   𝑉(π‘₯,𝑦,𝑧)

Proof of Theorem joinfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 joinfval.j . . 3 ∨ = (joinβ€˜πΎ)
3 fvex 6904 . . . . . . 7 (Baseβ€˜πΎ) ∈ V
4 moeq 3703 . . . . . . . 8 βˆƒ*𝑧 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦})
54a1i 11 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ βˆƒ*𝑧 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))
6 eqid 2732 . . . . . . 7 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}
73, 3, 5, 6oprabex 7965 . . . . . 6 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))} ∈ V
87a1i 11 . . . . 5 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))} ∈ V)
9 joinfval.u . . . . . . . . . . . 12 π‘ˆ = (lubβ€˜πΎ)
109lubfun 18307 . . . . . . . . . . 11 Fun π‘ˆ
11 funbrfv2b 6949 . . . . . . . . . . 11 (Fun π‘ˆ β†’ ({π‘₯, 𝑦}π‘ˆπ‘§ ↔ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{π‘₯, 𝑦}) = 𝑧)))
1210, 11ax-mp 5 . . . . . . . . . 10 ({π‘₯, 𝑦}π‘ˆπ‘§ ↔ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{π‘₯, 𝑦}) = 𝑧))
13 eqid 2732 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2732 . . . . . . . . . . . . . 14 (leβ€˜πΎ) = (leβ€˜πΎ)
15 simpl 483 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom π‘ˆ) β†’ 𝐾 ∈ V)
16 simpr 485 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom π‘ˆ) β†’ {π‘₯, 𝑦} ∈ dom π‘ˆ)
1713, 14, 9, 15, 16lubelss 18309 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom π‘ˆ) β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
1817ex 413 . . . . . . . . . . . 12 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ)))
19 vex 3478 . . . . . . . . . . . . 13 π‘₯ ∈ V
20 vex 3478 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4823 . . . . . . . . . . . 12 ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ↔ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
2218, 21imbitrrdi 251 . . . . . . . . . . 11 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ β†’ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))))
23 eqcom 2739 . . . . . . . . . . . 12 ((π‘ˆβ€˜{π‘₯, 𝑦}) = 𝑧 ↔ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))
2423biimpi 215 . . . . . . . . . . 11 ((π‘ˆβ€˜{π‘₯, 𝑦}) = 𝑧 β†’ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))
2522, 24anim12d1 610 . . . . . . . . . 10 (𝐾 ∈ V β†’ (({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{π‘₯, 𝑦}) = 𝑧) β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))))
2612, 25biimtrid 241 . . . . . . . . 9 (𝐾 ∈ V β†’ ({π‘₯, 𝑦}π‘ˆπ‘§ β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))))
2726alrimiv 1930 . . . . . . . 8 (𝐾 ∈ V β†’ βˆ€π‘§({π‘₯, 𝑦}π‘ˆπ‘§ β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))))
2827alrimiv 1930 . . . . . . 7 (𝐾 ∈ V β†’ βˆ€π‘¦βˆ€π‘§({π‘₯, 𝑦}π‘ˆπ‘§ β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))))
2928alrimiv 1930 . . . . . 6 (𝐾 ∈ V β†’ βˆ€π‘₯βˆ€π‘¦βˆ€π‘§({π‘₯, 𝑦}π‘ˆπ‘§ β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))))
30 ssoprab2 7479 . . . . . 6 (βˆ€π‘₯βˆ€π‘¦βˆ€π‘§({π‘₯, 𝑦}π‘ˆπ‘§ β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))) β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§} βŠ† {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))})
3129, 30syl 17 . . . . 5 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§} βŠ† {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))})
328, 31ssexd 5324 . . . 4 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§} ∈ V)
33 fveq2 6891 . . . . . . . 8 (𝑝 = 𝐾 β†’ (lubβ€˜π‘) = (lubβ€˜πΎ))
3433, 9eqtr4di 2790 . . . . . . 7 (𝑝 = 𝐾 β†’ (lubβ€˜π‘) = π‘ˆ)
3534breqd 5159 . . . . . 6 (𝑝 = 𝐾 β†’ ({π‘₯, 𝑦} (lubβ€˜π‘)𝑧 ↔ {π‘₯, 𝑦}π‘ˆπ‘§))
3635oprabbidv 7477 . . . . 5 (𝑝 = 𝐾 β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (lubβ€˜π‘)𝑧} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
37 df-join 18303 . . . . 5 join = (𝑝 ∈ V ↦ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (lubβ€˜π‘)𝑧})
3836, 37fvmptg 6996 . . . 4 ((𝐾 ∈ V ∧ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§} ∈ V) β†’ (joinβ€˜πΎ) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
3932, 38mpdan 685 . . 3 (𝐾 ∈ V β†’ (joinβ€˜πΎ) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
402, 39eqtrid 2784 . 2 (𝐾 ∈ V β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
411, 40syl 17 1 (𝐾 ∈ 𝑉 β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106  βˆƒ*wmo 2532  Vcvv 3474   βŠ† wss 3948  {cpr 4630   class class class wbr 5148  dom cdm 5676  Fun wfun 6537  β€˜cfv 6543  {coprab 7412  Basecbs 17146  lecple 17206  lubclub 18264  joincjn 18266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-oprab 7415  df-lub 18301  df-join 18303
This theorem is referenced by:  joinfval2  18329  join0  18360  odujoin  18363  odumeet  18365
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