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Theorem joinval 16828
 Description: Join value. Since both sides evaluate to ∅ when they don't exist, for convenience we drop the {𝑋, 𝑌} ∈ dom 𝑈 requirement. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
joindef.u 𝑈 = (lub‘𝐾)
joindef.j = (join‘𝐾)
joindef.k (𝜑𝐾𝑉)
joindef.x (𝜑𝑋𝑊)
joindef.y (𝜑𝑌𝑍)
Assertion
Ref Expression
joinval (𝜑 → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))

Proof of Theorem joinval
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 joindef.k . . . . . 6 (𝜑𝐾𝑉)
2 joindef.u . . . . . . 7 𝑈 = (lub‘𝐾)
3 joindef.j . . . . . . 7 = (join‘𝐾)
42, 3joinfval2 16825 . . . . . 6 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))})
51, 4syl 17 . . . . 5 (𝜑 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))})
65oveqd 6566 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
76adantr 480 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
8 simpr 476 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → {𝑋, 𝑌} ∈ dom 𝑈)
9 eqidd 2611 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))
10 joindef.x . . . . . 6 (𝜑𝑋𝑊)
11 joindef.y . . . . . 6 (𝜑𝑌𝑍)
12 fvex 6113 . . . . . . 7 (𝑈‘{𝑋, 𝑌}) ∈ V
1312a1i 11 . . . . . 6 (𝜑 → (𝑈‘{𝑋, 𝑌}) ∈ V)
14 preq12 4214 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
1514eleq1d 2672 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
16153adant3 1074 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
17 simp3 1056 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → 𝑧 = (𝑈‘{𝑋, 𝑌}))
1814fveq2d 6107 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
19183adant3 1074 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
2017, 19eqeq12d 2625 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑧 = (𝑈‘{𝑥, 𝑦}) ↔ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})))
2116, 20anbi12d 743 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))))
22 moeq 3349 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
2322moani 2513 . . . . . . 7 ∃*𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))
24 eqid 2610 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}
2521, 23, 24ovigg 6679 . . . . . 6 ((𝑋𝑊𝑌𝑍 ∧ (𝑈‘{𝑋, 𝑌}) ∈ V) → (({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌})))
2610, 11, 13, 25syl3anc 1318 . . . . 5 (𝜑 → (({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌})))
2726adantr 480 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌})))
288, 9, 27mp2and 711 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌}))
297, 28eqtrd 2644 . 2 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
302, 3, 1, 10, 11joindef 16827 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))
3130notbid 307 . . . . 5 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ↔ ¬ {𝑋, 𝑌} ∈ dom 𝑈))
32 df-ov 6552 . . . . . 6 (𝑋 𝑌) = ( ‘⟨𝑋, 𝑌⟩)
33 ndmfv 6128 . . . . . 6 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → ( ‘⟨𝑋, 𝑌⟩) = ∅)
3432, 33syl5eq 2656 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → (𝑋 𝑌) = ∅)
3531, 34syl6bir 243 . . . 4 (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑋 𝑌) = ∅))
3635imp 444 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = ∅)
37 ndmfv 6128 . . . 4 (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑈‘{𝑋, 𝑌}) = ∅)
3837adantl 481 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = ∅)
3936, 38eqtr4d 2647 . 2 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
4029, 39pm2.61dan 828 1 (𝜑 → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cop 4131  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  {coprab 6550  lubclub 16765  joincjn 16767 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-lub 16797  df-join 16799 This theorem is referenced by:  joincl  16829  joinval2  16832  joincomALT  16852  lubsn  16917
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