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Theorem joinval 18265
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 18262 . . . . . 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 7373 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
76adantr 481 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
8 simpr 485 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → {𝑋, 𝑌} ∈ dom 𝑈)
9 eqidd 2737 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))
10 joindef.x . . . . . 6 (𝜑𝑋𝑊)
11 joindef.y . . . . . 6 (𝜑𝑌𝑍)
12 fvexd 6857 . . . . . 6 (𝜑 → (𝑈‘{𝑋, 𝑌}) ∈ V)
13 preq12 4696 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
1413eleq1d 2822 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15143adant3 1132 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
16 simp3 1138 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → 𝑧 = (𝑈‘{𝑋, 𝑌}))
1713fveq2d 6846 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
18173adant3 1132 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
1916, 18eqeq12d 2752 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑧 = (𝑈‘{𝑥, 𝑦}) ↔ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})))
2015, 19anbi12d 631 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦})) ↔ ({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))))
21 moeq 3665 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
2221moani 2551 . . . . . . 7 ∃*𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))
23 eqid 2736 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}
2420, 22, 23ovigg 7499 . . . . . 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})))
2510, 11, 12, 24syl3anc 1371 . . . . 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})))
2625adantr 481 . . . 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})))
278, 9, 26mp2and 697 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌}))
287, 27eqtrd 2776 . 2 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
292, 3, 1, 10, 11joindef 18264 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))
3029notbid 317 . . . . 5 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ↔ ¬ {𝑋, 𝑌} ∈ dom 𝑈))
31 df-ov 7359 . . . . . 6 (𝑋 𝑌) = ( ‘⟨𝑋, 𝑌⟩)
32 ndmfv 6877 . . . . . 6 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → ( ‘⟨𝑋, 𝑌⟩) = ∅)
3331, 32eqtrid 2788 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → (𝑋 𝑌) = ∅)
3430, 33syl6bir 253 . . . 4 (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑋 𝑌) = ∅))
3534imp 407 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = ∅)
36 ndmfv 6877 . . . 4 (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑈‘{𝑋, 𝑌}) = ∅)
3736adantl 482 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = ∅)
3835, 37eqtr4d 2779 . 2 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
3928, 38pm2.61dan 811 1 (𝜑 → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  c0 4282  {cpr 4588  cop 4592  dom cdm 5633  cfv 6496  (class class class)co 7356  {coprab 7357  lubclub 18197  joincjn 18199
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-ov 7359  df-oprab 7360  df-lub 18234  df-join 18236
This theorem is referenced by:  joincl  18266  joinval2  18269  joincomALT  18289  lubsn  18370  posjidm  46976  toplatjoin  46998
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