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Theorem djuun 9918
Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.)
Assertion
Ref Expression
djuun ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) = (𝐴 βŠ” 𝐡)
Proof of Theorem djuun
Dummy variables π‘₯ 𝑦 𝑒 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elun 4148 . . . 4 (π‘₯ ∈ ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) ↔ (π‘₯ ∈ (inl β€œ 𝐴) ∨ π‘₯ ∈ (inr β€œ 𝐡)))
2 djulf1o 9904 . . . . . . . . . . 11 inl:V–1-1-ontoβ†’({βˆ…} Γ— V)
3 f1ofn 6832 . . . . . . . . . . 11 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ inl Fn V)
42, 3ax-mp 5 . . . . . . . . . 10 inl Fn V
5 ssv 4006 . . . . . . . . . 10 𝐴 βŠ† V
6 fvelimab 6962 . . . . . . . . . 10 ((inl Fn V ∧ 𝐴 βŠ† V) β†’ (π‘₯ ∈ (inl β€œ 𝐴) ↔ βˆƒπ‘’ ∈ 𝐴 (inlβ€˜π‘’) = π‘₯))
74, 5, 6mp2an 691 . . . . . . . . 9 (π‘₯ ∈ (inl β€œ 𝐴) ↔ βˆƒπ‘’ ∈ 𝐴 (inlβ€˜π‘’) = π‘₯)
87biimpi 215 . . . . . . . 8 (π‘₯ ∈ (inl β€œ 𝐴) β†’ βˆƒπ‘’ ∈ 𝐴 (inlβ€˜π‘’) = π‘₯)
9 simprr 772 . . . . . . . . 9 ((π‘₯ ∈ (inl β€œ 𝐴) ∧ (𝑒 ∈ 𝐴 ∧ (inlβ€˜π‘’) = π‘₯)) β†’ (inlβ€˜π‘’) = π‘₯)
10 vex 3479 . . . . . . . . . . 11 𝑒 ∈ V
11 opex 5464 . . . . . . . . . . 11 βŸ¨βˆ…, π‘’βŸ© ∈ V
12 opeq2 4874 . . . . . . . . . . . 12 (𝑧 = 𝑒 β†’ βŸ¨βˆ…, π‘§βŸ© = βŸ¨βˆ…, π‘’βŸ©)
13 df-inl 9894 . . . . . . . . . . . 12 inl = (𝑧 ∈ V ↦ βŸ¨βˆ…, π‘§βŸ©)
1412, 13fvmptg 6994 . . . . . . . . . . 11 ((𝑒 ∈ V ∧ βŸ¨βˆ…, π‘’βŸ© ∈ V) β†’ (inlβ€˜π‘’) = βŸ¨βˆ…, π‘’βŸ©)
1510, 11, 14mp2an 691 . . . . . . . . . 10 (inlβ€˜π‘’) = βŸ¨βˆ…, π‘’βŸ©
16 0ex 5307 . . . . . . . . . . . . 13 βˆ… ∈ V
1716snid 4664 . . . . . . . . . . . 12 βˆ… ∈ {βˆ…}
18 opelxpi 5713 . . . . . . . . . . . 12 ((βˆ… ∈ {βˆ…} ∧ 𝑒 ∈ 𝐴) β†’ βŸ¨βˆ…, π‘’βŸ© ∈ ({βˆ…} Γ— 𝐴))
1917, 18mpan 689 . . . . . . . . . . 11 (𝑒 ∈ 𝐴 β†’ βŸ¨βˆ…, π‘’βŸ© ∈ ({βˆ…} Γ— 𝐴))
2019ad2antrl 727 . . . . . . . . . 10 ((π‘₯ ∈ (inl β€œ 𝐴) ∧ (𝑒 ∈ 𝐴 ∧ (inlβ€˜π‘’) = π‘₯)) β†’ βŸ¨βˆ…, π‘’βŸ© ∈ ({βˆ…} Γ— 𝐴))
2115, 20eqeltrid 2838 . . . . . . . . 9 ((π‘₯ ∈ (inl β€œ 𝐴) ∧ (𝑒 ∈ 𝐴 ∧ (inlβ€˜π‘’) = π‘₯)) β†’ (inlβ€˜π‘’) ∈ ({βˆ…} Γ— 𝐴))
229, 21eqeltrrd 2835 . . . . . . . 8 ((π‘₯ ∈ (inl β€œ 𝐴) ∧ (𝑒 ∈ 𝐴 ∧ (inlβ€˜π‘’) = π‘₯)) β†’ π‘₯ ∈ ({βˆ…} Γ— 𝐴))
238, 22rexlimddv 3162 . . . . . . 7 (π‘₯ ∈ (inl β€œ 𝐴) β†’ π‘₯ ∈ ({βˆ…} Γ— 𝐴))
24 elun1 4176 . . . . . . 7 (π‘₯ ∈ ({βˆ…} Γ— 𝐴) β†’ π‘₯ ∈ (({βˆ…} Γ— 𝐴) βˆͺ ({1o} Γ— 𝐡)))
2523, 24syl 17 . . . . . 6 (π‘₯ ∈ (inl β€œ 𝐴) β†’ π‘₯ ∈ (({βˆ…} Γ— 𝐴) βˆͺ ({1o} Γ— 𝐡)))
26 df-dju 9893 . . . . . 6 (𝐴 βŠ” 𝐡) = (({βˆ…} Γ— 𝐴) βˆͺ ({1o} Γ— 𝐡))
2725, 26eleqtrrdi 2845 . . . . 5 (π‘₯ ∈ (inl β€œ 𝐴) β†’ π‘₯ ∈ (𝐴 βŠ” 𝐡))
28 djurf1o 9905 . . . . . . . . . . 11 inr:V–1-1-ontoβ†’({1o} Γ— V)
29 f1ofn 6832 . . . . . . . . . . 11 (inr:V–1-1-ontoβ†’({1o} Γ— V) β†’ inr Fn V)
3028, 29ax-mp 5 . . . . . . . . . 10 inr Fn V
31 ssv 4006 . . . . . . . . . 10 𝐡 βŠ† V
32 fvelimab 6962 . . . . . . . . . 10 ((inr Fn V ∧ 𝐡 βŠ† V) β†’ (π‘₯ ∈ (inr β€œ 𝐡) ↔ βˆƒπ‘’ ∈ 𝐡 (inrβ€˜π‘’) = π‘₯))
3330, 31, 32mp2an 691 . . . . . . . . 9 (π‘₯ ∈ (inr β€œ 𝐡) ↔ βˆƒπ‘’ ∈ 𝐡 (inrβ€˜π‘’) = π‘₯)
3433biimpi 215 . . . . . . . 8 (π‘₯ ∈ (inr β€œ 𝐡) β†’ βˆƒπ‘’ ∈ 𝐡 (inrβ€˜π‘’) = π‘₯)
35 simprr 772 . . . . . . . . 9 ((π‘₯ ∈ (inr β€œ 𝐡) ∧ (𝑒 ∈ 𝐡 ∧ (inrβ€˜π‘’) = π‘₯)) β†’ (inrβ€˜π‘’) = π‘₯)
36 opex 5464 . . . . . . . . . . 11 ⟨1o, π‘’βŸ© ∈ V
37 opeq2 4874 . . . . . . . . . . . 12 (𝑧 = 𝑒 β†’ ⟨1o, π‘§βŸ© = ⟨1o, π‘’βŸ©)
38 df-inr 9895 . . . . . . . . . . . 12 inr = (𝑧 ∈ V ↦ ⟨1o, π‘§βŸ©)
3937, 38fvmptg 6994 . . . . . . . . . . 11 ((𝑒 ∈ V ∧ ⟨1o, π‘’βŸ© ∈ V) β†’ (inrβ€˜π‘’) = ⟨1o, π‘’βŸ©)
4010, 36, 39mp2an 691 . . . . . . . . . 10 (inrβ€˜π‘’) = ⟨1o, π‘’βŸ©
41 1oex 8473 . . . . . . . . . . . . 13 1o ∈ V
4241snid 4664 . . . . . . . . . . . 12 1o ∈ {1o}
43 opelxpi 5713 . . . . . . . . . . . 12 ((1o ∈ {1o} ∧ 𝑒 ∈ 𝐡) β†’ ⟨1o, π‘’βŸ© ∈ ({1o} Γ— 𝐡))
4442, 43mpan 689 . . . . . . . . . . 11 (𝑒 ∈ 𝐡 β†’ ⟨1o, π‘’βŸ© ∈ ({1o} Γ— 𝐡))
4544ad2antrl 727 . . . . . . . . . 10 ((π‘₯ ∈ (inr β€œ 𝐡) ∧ (𝑒 ∈ 𝐡 ∧ (inrβ€˜π‘’) = π‘₯)) β†’ ⟨1o, π‘’βŸ© ∈ ({1o} Γ— 𝐡))
4640, 45eqeltrid 2838 . . . . . . . . 9 ((π‘₯ ∈ (inr β€œ 𝐡) ∧ (𝑒 ∈ 𝐡 ∧ (inrβ€˜π‘’) = π‘₯)) β†’ (inrβ€˜π‘’) ∈ ({1o} Γ— 𝐡))
4735, 46eqeltrrd 2835 . . . . . . . 8 ((π‘₯ ∈ (inr β€œ 𝐡) ∧ (𝑒 ∈ 𝐡 ∧ (inrβ€˜π‘’) = π‘₯)) β†’ π‘₯ ∈ ({1o} Γ— 𝐡))
4834, 47rexlimddv 3162 . . . . . . 7 (π‘₯ ∈ (inr β€œ 𝐡) β†’ π‘₯ ∈ ({1o} Γ— 𝐡))
49 elun2 4177 . . . . . . 7 (π‘₯ ∈ ({1o} Γ— 𝐡) β†’ π‘₯ ∈ (({βˆ…} Γ— 𝐴) βˆͺ ({1o} Γ— 𝐡)))
5048, 49syl 17 . . . . . 6 (π‘₯ ∈ (inr β€œ 𝐡) β†’ π‘₯ ∈ (({βˆ…} Γ— 𝐴) βˆͺ ({1o} Γ— 𝐡)))
5150, 26eleqtrrdi 2845 . . . . 5 (π‘₯ ∈ (inr β€œ 𝐡) β†’ π‘₯ ∈ (𝐴 βŠ” 𝐡))
5227, 51jaoi 856 . . . 4 ((π‘₯ ∈ (inl β€œ 𝐴) ∨ π‘₯ ∈ (inr β€œ 𝐡)) β†’ π‘₯ ∈ (𝐴 βŠ” 𝐡))
531, 52sylbi 216 . . 3 (π‘₯ ∈ ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) β†’ π‘₯ ∈ (𝐴 βŠ” 𝐡))
5453ssriv 3986 . 2 ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) βŠ† (𝐴 βŠ” 𝐡)
55 djur 9911 . . . . 5 (π‘₯ ∈ (𝐴 βŠ” 𝐡) β†’ (βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (inlβ€˜π‘¦) ∨ βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (inrβ€˜π‘¦)))
56 vex 3479 . . . . . . . . . 10 𝑦 ∈ V
57 f1odm 6835 . . . . . . . . . . 11 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ dom inl = V)
582, 57ax-mp 5 . . . . . . . . . 10 dom inl = V
5956, 58eleqtrri 2833 . . . . . . . . 9 𝑦 ∈ dom inl
60 simpl 484 . . . . . . . . 9 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ 𝑦 ∈ 𝐴)
6113funmpt2 6585 . . . . . . . . . 10 Fun inl
62 funfvima 7229 . . . . . . . . . 10 ((Fun inl ∧ 𝑦 ∈ dom inl) β†’ (𝑦 ∈ 𝐴 β†’ (inlβ€˜π‘¦) ∈ (inl β€œ 𝐴)))
6361, 62mpan 689 . . . . . . . . 9 (𝑦 ∈ dom inl β†’ (𝑦 ∈ 𝐴 β†’ (inlβ€˜π‘¦) ∈ (inl β€œ 𝐴)))
6459, 60, 63mpsyl 68 . . . . . . . 8 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ (inlβ€˜π‘¦) ∈ (inl β€œ 𝐴))
65 eleq1 2822 . . . . . . . . 9 (π‘₯ = (inlβ€˜π‘¦) β†’ (π‘₯ ∈ (inl β€œ 𝐴) ↔ (inlβ€˜π‘¦) ∈ (inl β€œ 𝐴)))
6665adantl 483 . . . . . . . 8 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ (π‘₯ ∈ (inl β€œ 𝐴) ↔ (inlβ€˜π‘¦) ∈ (inl β€œ 𝐴)))
6764, 66mpbird 257 . . . . . . 7 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ π‘₯ ∈ (inl β€œ 𝐴))
6867rexlimiva 3148 . . . . . 6 (βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (inlβ€˜π‘¦) β†’ π‘₯ ∈ (inl β€œ 𝐴))
69 f1odm 6835 . . . . . . . . . . 11 (inr:V–1-1-ontoβ†’({1o} Γ— V) β†’ dom inr = V)
7028, 69ax-mp 5 . . . . . . . . . 10 dom inr = V
7156, 70eleqtrri 2833 . . . . . . . . 9 𝑦 ∈ dom inr
72 simpl 484 . . . . . . . . 9 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ 𝑦 ∈ 𝐡)
73 f1ofun 6833 . . . . . . . . . . 11 (inr:V–1-1-ontoβ†’({1o} Γ— V) β†’ Fun inr)
7428, 73ax-mp 5 . . . . . . . . . 10 Fun inr
75 funfvima 7229 . . . . . . . . . 10 ((Fun inr ∧ 𝑦 ∈ dom inr) β†’ (𝑦 ∈ 𝐡 β†’ (inrβ€˜π‘¦) ∈ (inr β€œ 𝐡)))
7674, 75mpan 689 . . . . . . . . 9 (𝑦 ∈ dom inr β†’ (𝑦 ∈ 𝐡 β†’ (inrβ€˜π‘¦) ∈ (inr β€œ 𝐡)))
7771, 72, 76mpsyl 68 . . . . . . . 8 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ (inrβ€˜π‘¦) ∈ (inr β€œ 𝐡))
78 eleq1 2822 . . . . . . . . 9 (π‘₯ = (inrβ€˜π‘¦) β†’ (π‘₯ ∈ (inr β€œ 𝐡) ↔ (inrβ€˜π‘¦) ∈ (inr β€œ 𝐡)))
7978adantl 483 . . . . . . . 8 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ (π‘₯ ∈ (inr β€œ 𝐡) ↔ (inrβ€˜π‘¦) ∈ (inr β€œ 𝐡)))
8077, 79mpbird 257 . . . . . . 7 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ π‘₯ ∈ (inr β€œ 𝐡))
8180rexlimiva 3148 . . . . . 6 (βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (inrβ€˜π‘¦) β†’ π‘₯ ∈ (inr β€œ 𝐡))
8268, 81orim12i 908 . . . . 5 ((βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (inlβ€˜π‘¦) ∨ βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (inrβ€˜π‘¦)) β†’ (π‘₯ ∈ (inl β€œ 𝐴) ∨ π‘₯ ∈ (inr β€œ 𝐡)))
8355, 82syl 17 . . . 4 (π‘₯ ∈ (𝐴 βŠ” 𝐡) β†’ (π‘₯ ∈ (inl β€œ 𝐴) ∨ π‘₯ ∈ (inr β€œ 𝐡)))
8483, 1sylibr 233 . . 3 (π‘₯ ∈ (𝐴 βŠ” 𝐡) β†’ π‘₯ ∈ ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)))
8584ssriv 3986 . 2 (𝐴 βŠ” 𝐡) βŠ† ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡))
8654, 85eqssi 3998 1 ((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) = (𝐴 βŠ” 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3946   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βŸ¨cop 4634   Γ— cxp 5674  dom cdm 5676   β€œ cima 5679  Fun wfun 6535   Fn wfn 6536  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  1oc1o 8456   βŠ” cdju 9890  inlcinl 9891  inrcinr 9892
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-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-om 7853  df-1st 7972  df-2nd 7973  df-1o 8463  df-dju 9893  df-inl 9894  df-inr 9895
This theorem is referenced by: (None)
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