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Theorem joinval 17209
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 17206 . . . . . 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 6809 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
76adantr 466 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌))
8 simpr 471 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → {𝑋, 𝑌} ∈ dom 𝑈)
9 eqidd 2772 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌}))
10 joindef.x . . . . . 6 (𝜑𝑋𝑊)
11 joindef.y . . . . . 6 (𝜑𝑌𝑍)
12 fvexd 6343 . . . . . 6 (𝜑 → (𝑈‘{𝑋, 𝑌}) ∈ V)
13 preq12 4406 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
1413eleq1d 2835 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
15143adant3 1126 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
16 simp3 1132 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → 𝑧 = (𝑈‘{𝑋, 𝑌}))
1713fveq2d 6334 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
18173adant3 1126 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑈‘{𝑥, 𝑦}) = (𝑈‘{𝑋, 𝑌}))
1916, 18eqeq12d 2786 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝑈‘{𝑋, 𝑌})) → (𝑧 = (𝑈‘{𝑥, 𝑦}) ↔ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑋, 𝑌})))
2015, 19anbi12d 616 . . . . . . 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 3534 . . . . . . . 8 ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
2221moani 2674 . . . . . . 7 ∃*𝑧({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))
23 eqid 2771 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}
2420, 22, 23ovigg 6928 . . . . . 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 1476 . . . . 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 466 . . . 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 679 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))}𝑌) = (𝑈‘{𝑋, 𝑌}))
287, 27eqtrd 2805 . 2 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
292, 3, 1, 10, 11joindef 17208 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))
3029notbid 307 . . . . 5 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ↔ ¬ {𝑋, 𝑌} ∈ dom 𝑈))
31 df-ov 6795 . . . . . 6 (𝑋 𝑌) = ( ‘⟨𝑋, 𝑌⟩)
32 ndmfv 6358 . . . . . 6 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → ( ‘⟨𝑋, 𝑌⟩) = ∅)
3331, 32syl5eq 2817 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → (𝑋 𝑌) = ∅)
3430, 33syl6bir 244 . . . 4 (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑋 𝑌) = ∅))
3534imp 393 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = ∅)
36 ndmfv 6358 . . . 4 (¬ {𝑋, 𝑌} ∈ dom 𝑈 → (𝑈‘{𝑋, 𝑌}) = ∅)
3736adantl 467 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑈‘{𝑋, 𝑌}) = ∅)
3835, 37eqtr4d 2808 . 2 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝑈) → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
3928, 38pm2.61dan 814 1 (𝜑 → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  {cpr 4318  cop 4322  dom cdm 5249  cfv 6029  (class class class)co 6792  {coprab 6793  lubclub 17146  joincjn 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6753  df-ov 6795  df-oprab 6796  df-lub 17178  df-join 17180
This theorem is referenced by:  joincl  17210  joinval2  17213  joincomALT  17233  lubsn  17298
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