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Theorem joinval 18326
Description: Join value. Since both sides evaluate to βˆ… when they don't exist, for convenience we drop the {𝑋, π‘Œ} ∈ dom π‘ˆ requirement. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
joindef.u π‘ˆ = (lubβ€˜πΎ)
joindef.j ∨ = (joinβ€˜πΎ)
joindef.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
joindef.x (πœ‘ β†’ 𝑋 ∈ π‘Š)
joindef.y (πœ‘ β†’ π‘Œ ∈ 𝑍)
Assertion
Ref Expression
joinval (πœ‘ β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))

Proof of Theorem joinval
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 joindef.k . . . . . 6 (πœ‘ β†’ 𝐾 ∈ 𝑉)
2 joindef.u . . . . . . 7 π‘ˆ = (lubβ€˜πΎ)
3 joindef.j . . . . . . 7 ∨ = (joinβ€˜πΎ)
42, 3joinfval2 18323 . . . . . 6 (𝐾 ∈ 𝑉 β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))})
51, 4syl 17 . . . . 5 (πœ‘ β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))})
65oveqd 7422 . . . 4 (πœ‘ β†’ (𝑋 ∨ π‘Œ) = (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}π‘Œ))
76adantr 481 . . 3 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}π‘Œ))
8 simpr 485 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ {𝑋, π‘Œ} ∈ dom π‘ˆ)
9 eqidd 2733 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) = (π‘ˆβ€˜{𝑋, π‘Œ}))
10 joindef.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ π‘Š)
11 joindef.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑍)
12 fvexd 6903 . . . . . 6 (πœ‘ β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) ∈ V)
13 preq12 4738 . . . . . . . . . 10 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ {π‘₯, 𝑦} = {𝑋, π‘Œ})
1413eleq1d 2818 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
15143adant3 1132 . . . . . . . 8 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
16 simp3 1138 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ}))
1713fveq2d 6892 . . . . . . . . . 10 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘ˆβ€˜{π‘₯, 𝑦}) = (π‘ˆβ€˜{𝑋, π‘Œ}))
18173adant3 1132 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (π‘ˆβ€˜{π‘₯, 𝑦}) = (π‘ˆβ€˜{𝑋, π‘Œ}))
1916, 18eqeq12d 2748 . . . . . . . 8 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}) ↔ (π‘ˆβ€˜{𝑋, π‘Œ}) = (π‘ˆβ€˜{𝑋, π‘Œ})))
2015, 19anbi12d 631 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦})) ↔ ({𝑋, π‘Œ} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{𝑋, π‘Œ}) = (π‘ˆβ€˜{𝑋, π‘Œ}))))
21 moeq 3702 . . . . . . . 8 βˆƒ*𝑧 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦})
2221moani 2547 . . . . . . 7 βˆƒ*𝑧({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))
23 eqid 2732 . . . . . . 7 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}
2420, 22, 23ovigg 7549 . . . . . 6 ((𝑋 ∈ π‘Š ∧ π‘Œ ∈ 𝑍 ∧ (π‘ˆβ€˜{𝑋, π‘Œ}) ∈ V) β†’ (({𝑋, π‘Œ} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{𝑋, π‘Œ}) = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ})))
2510, 11, 12, 24syl3anc 1371 . . . . 5 (πœ‘ β†’ (({𝑋, π‘Œ} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{𝑋, π‘Œ}) = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ})))
2625adantr 481 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (({𝑋, π‘Œ} ∈ dom π‘ˆ ∧ (π‘ˆβ€˜{𝑋, π‘Œ}) = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ})))
278, 9, 26mp2and 697 . . 3 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))}π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
287, 27eqtrd 2772 . 2 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
292, 3, 1, 10, 11joindef 18325 . . . . . 6 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
3029notbid 317 . . . . 5 (πœ‘ β†’ (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ))
31 df-ov 7408 . . . . . 6 (𝑋 ∨ π‘Œ) = ( ∨ β€˜βŸ¨π‘‹, π‘ŒβŸ©)
32 ndmfv 6923 . . . . . 6 (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ β†’ ( ∨ β€˜βŸ¨π‘‹, π‘ŒβŸ©) = βˆ…)
3331, 32eqtrid 2784 . . . . 5 (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ β†’ (𝑋 ∨ π‘Œ) = βˆ…)
3430, 33syl6bir 253 . . . 4 (πœ‘ β†’ (Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ β†’ (𝑋 ∨ π‘Œ) = βˆ…))
3534imp 407 . . 3 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = βˆ…)
36 ndmfv 6923 . . . 4 (Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) = βˆ…)
3736adantl 482 . . 3 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) = βˆ…)
3835, 37eqtr4d 2775 . 2 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
3928, 38pm2.61dan 811 1 (πœ‘ β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βˆ…c0 4321  {cpr 4629  βŸ¨cop 4633  dom cdm 5675  β€˜cfv 6540  (class class class)co 7405  {coprab 7406  lubclub 18258  joincjn 18260
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-lub 18295  df-join 18297
This theorem is referenced by:  joincl  18327  joinval2  18330  joincomALT  18350  lubsn  18431  posjidm  47558  toplatjoin  47580
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