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Theorem joinfval 18267
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 18268 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 3462 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 joinfval.j . . 3 ∨ = (joinβ€˜πΎ)
3 fvex 6856 . . . . . . 7 (Baseβ€˜πΎ) ∈ V
4 moeq 3666 . . . . . . . 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 2733 . . . . . . 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 7910 . . . . . 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 18246 . . . . . . . . . . 11 Fun π‘ˆ
11 funbrfv2b 6901 . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2733 . . . . . . . . . . . . . 14 (leβ€˜πΎ) = (leβ€˜πΎ)
15 simpl 484 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom π‘ˆ) β†’ 𝐾 ∈ V)
16 simpr 486 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom π‘ˆ) β†’ {π‘₯, 𝑦} ∈ dom π‘ˆ)
1713, 14, 9, 15, 16lubelss 18248 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom π‘ˆ) β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
1817ex 414 . . . . . . . . . . . 12 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ)))
19 vex 3448 . . . . . . . . . . . . 13 π‘₯ ∈ V
20 vex 3448 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4781 . . . . . . . . . . . 12 ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ↔ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
2218, 21syl6ibr 252 . . . . . . . . . . 11 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ β†’ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))))
23 eqcom 2740 . . . . . . . . . . . 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 611 . . . . . . . . . 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 1931 . . . . . . . 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 1931 . . . . . . 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 1931 . . . . . 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 7426 . . . . . 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 5282 . . . 4 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§} ∈ V)
33 fveq2 6843 . . . . . . . 8 (𝑝 = 𝐾 β†’ (lubβ€˜π‘) = (lubβ€˜πΎ))
3433, 9eqtr4di 2791 . . . . . . 7 (𝑝 = 𝐾 β†’ (lubβ€˜π‘) = π‘ˆ)
3534breqd 5117 . . . . . 6 (𝑝 = 𝐾 β†’ ({π‘₯, 𝑦} (lubβ€˜π‘)𝑧 ↔ {π‘₯, 𝑦}π‘ˆπ‘§))
3635oprabbidv 7424 . . . . 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 18242 . . . . 5 join = (𝑝 ∈ V ↦ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (lubβ€˜π‘)𝑧})
3836, 37fvmptg 6947 . . . 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 686 . . 3 (𝐾 ∈ V β†’ (joinβ€˜πΎ) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
402, 39eqtrid 2785 . 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 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆƒ*wmo 2533  Vcvv 3444   βŠ† wss 3911  {cpr 4589   class class class wbr 5106  dom cdm 5634  Fun wfun 6491  β€˜cfv 6497  {coprab 7359  Basecbs 17088  lecple 17145  lubclub 18203  joincjn 18205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-oprab 7362  df-lub 18240  df-join 18242
This theorem is referenced by:  joinfval2  18268  join0  18299  odujoin  18302  odumeet  18304
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