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Theorem joinval 17611
 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 17608 . . . . . 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 7156 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
76adantr 484 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
8 simpr 488 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → {𝑋, 𝑌} ∈ dom 𝑈)
9 eqidd 2802 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))
10 joindef.x . . . . . 6 (𝜑𝑋𝑊)
11 joindef.y . . . . . 6 (𝜑𝑌𝑍)
12 fvexd 6664 . . . . . 6 (𝜑 → (𝑈‘{𝑋, 𝑌}) ∈ V)
13 preq12 4634 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
1413eleq1d 2877 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15143adant3 1129 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
16 simp3 1135 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → 𝑧 = (𝑈‘{𝑋, 𝑌}))
1713fveq2d 6653 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
18173adant3 1129 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
1916, 18eqeq12d 2817 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑧 = (𝑈‘{𝑥, 𝑦}) ↔ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})))
2015, 19anbi12d 633 . . . . . . 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 3649 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
2221moani 2615 . . . . . . 7 ∃*𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))
23 eqid 2801 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}
2420, 22, 23ovigg 7278 . . . . . 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 1368 . . . . 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 484 . . . 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 698 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌}))
287, 27eqtrd 2836 . 2 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
292, 3, 1, 10, 11joindef 17610 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))
3029notbid 321 . . . . 5 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ↔ ¬ {𝑋, 𝑌} ∈ dom 𝑈))
31 df-ov 7142 . . . . . 6 (𝑋 𝑌) = ( ‘⟨𝑋, 𝑌⟩)
32 ndmfv 6679 . . . . . 6 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → ( ‘⟨𝑋, 𝑌⟩) = ∅)
3331, 32syl5eq 2848 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → (𝑋 𝑌) = ∅)
3430, 33syl6bir 257 . . . 4 (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑋 𝑌) = ∅))
3534imp 410 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = ∅)
36 ndmfv 6679 . . . 4 (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑈‘{𝑋, 𝑌}) = ∅)
3736adantl 485 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = ∅)
3835, 37eqtr4d 2839 . 2 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
3928, 38pm2.61dan 812 1 (𝜑 → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ∅c0 4246  {cpr 4530  ⟨cop 4534  dom cdm 5523  ‘cfv 6328  (class class class)co 7139  {coprab 7140  lubclub 17548  joincjn 17550 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-lub 17580  df-join 17582 This theorem is referenced by:  joincl  17612  joinval2  17615  joincomALT  17635  lubsn  17700
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