MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  joinval Structured version   Visualization version   GIF version

Theorem joinval 18330
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 18327 . . . . . 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 7426 . . . 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 482 . . 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 486 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ {𝑋, π‘Œ} ∈ dom π‘ˆ)
9 eqidd 2734 . . . 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 6907 . . . . . 6 (πœ‘ β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) ∈ V)
13 preq12 4740 . . . . . . . . . 10 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ {π‘₯, 𝑦} = {𝑋, π‘Œ})
1413eleq1d 2819 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ({π‘₯, 𝑦} ∈ dom π‘ˆ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
15143adant3 1133 . . . . . . . 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 1139 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ}))
1713fveq2d 6896 . . . . . . . . . 10 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘ˆβ€˜{π‘₯, 𝑦}) = (π‘ˆβ€˜{𝑋, π‘Œ}))
18173adant3 1133 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (π‘ˆβ€˜{𝑋, π‘Œ})) β†’ (π‘ˆβ€˜{π‘₯, 𝑦}) = (π‘ˆβ€˜{𝑋, π‘Œ}))
1916, 18eqeq12d 2749 . . . . . . . 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 632 . . . . . . 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 3704 . . . . . . . 8 βˆƒ*𝑧 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦})
2221moani 2548 . . . . . . 7 βˆƒ*𝑧({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))
23 eqid 2733 . . . . . . 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 7553 . . . . . 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 1372 . . . . 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 482 . . . 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 698 . . 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 2773 . 2 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
292, 3, 1, 10, 11joindef 18329 . . . . . 6 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
3029notbid 318 . . . . 5 (πœ‘ β†’ (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ))
31 df-ov 7412 . . . . . 6 (𝑋 ∨ π‘Œ) = ( ∨ β€˜βŸ¨π‘‹, π‘ŒβŸ©)
32 ndmfv 6927 . . . . . 6 (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ β†’ ( ∨ β€˜βŸ¨π‘‹, π‘ŒβŸ©) = βˆ…)
3331, 32eqtrid 2785 . . . . 5 (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ β†’ (𝑋 ∨ π‘Œ) = βˆ…)
3430, 33syl6bir 254 . . . 4 (πœ‘ β†’ (Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ β†’ (𝑋 ∨ π‘Œ) = βˆ…))
3534imp 408 . . 3 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = βˆ…)
36 ndmfv 6927 . . . 4 (Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) = βˆ…)
3736adantl 483 . . 3 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (π‘ˆβ€˜{𝑋, π‘Œ}) = βˆ…)
3835, 37eqtr4d 2776 . 2 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom π‘ˆ) β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
3928, 38pm2.61dan 812 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475  βˆ…c0 4323  {cpr 4631  βŸ¨cop 4635  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  {coprab 7410  lubclub 18262  joincjn 18264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-lub 18299  df-join 18301
This theorem is referenced by:  joincl  18331  joinval2  18334  joincomALT  18354  lubsn  18435  posjidm  47605  toplatjoin  47627
  Copyright terms: Public domain W3C validator