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Theorem djuun 9343
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 “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)

Proof of Theorem djuun
Dummy variables 𝑥 𝑦 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elun 4079 . . . 4 (𝑥 ∈ ((inl “ 𝐴) ∪ (inr “ 𝐵)) ↔ (𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)))
2 djulf1o 9329 . . . . . . . . . . 11 inl:V–1-1-onto→({∅} × V)
3 f1ofn 6595 . . . . . . . . . . 11 (inl:V–1-1-onto→({∅} × V) → inl Fn V)
42, 3ax-mp 5 . . . . . . . . . 10 inl Fn V
5 ssv 3942 . . . . . . . . . 10 𝐴 ⊆ V
6 fvelimab 6716 . . . . . . . . . 10 ((inl Fn V ∧ 𝐴 ⊆ V) → (𝑥 ∈ (inl “ 𝐴) ↔ ∃𝑢𝐴 (inl‘𝑢) = 𝑥))
74, 5, 6mp2an 691 . . . . . . . . 9 (𝑥 ∈ (inl “ 𝐴) ↔ ∃𝑢𝐴 (inl‘𝑢) = 𝑥)
87biimpi 219 . . . . . . . 8 (𝑥 ∈ (inl “ 𝐴) → ∃𝑢𝐴 (inl‘𝑢) = 𝑥)
9 simprr 772 . . . . . . . . 9 ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢𝐴 ∧ (inl‘𝑢) = 𝑥)) → (inl‘𝑢) = 𝑥)
10 vex 3447 . . . . . . . . . . 11 𝑢 ∈ V
11 opex 5324 . . . . . . . . . . 11 ⟨∅, 𝑢⟩ ∈ V
12 opeq2 4768 . . . . . . . . . . . 12 (𝑧 = 𝑢 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑢⟩)
13 df-inl 9319 . . . . . . . . . . . 12 inl = (𝑧 ∈ V ↦ ⟨∅, 𝑧⟩)
1412, 13fvmptg 6747 . . . . . . . . . . 11 ((𝑢 ∈ V ∧ ⟨∅, 𝑢⟩ ∈ V) → (inl‘𝑢) = ⟨∅, 𝑢⟩)
1510, 11, 14mp2an 691 . . . . . . . . . 10 (inl‘𝑢) = ⟨∅, 𝑢
16 0ex 5178 . . . . . . . . . . . . 13 ∅ ∈ V
1716snid 4564 . . . . . . . . . . . 12 ∅ ∈ {∅}
18 opelxpi 5560 . . . . . . . . . . . 12 ((∅ ∈ {∅} ∧ 𝑢𝐴) → ⟨∅, 𝑢⟩ ∈ ({∅} × 𝐴))
1917, 18mpan 689 . . . . . . . . . . 11 (𝑢𝐴 → ⟨∅, 𝑢⟩ ∈ ({∅} × 𝐴))
2019ad2antrl 727 . . . . . . . . . 10 ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢𝐴 ∧ (inl‘𝑢) = 𝑥)) → ⟨∅, 𝑢⟩ ∈ ({∅} × 𝐴))
2115, 20eqeltrid 2897 . . . . . . . . 9 ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢𝐴 ∧ (inl‘𝑢) = 𝑥)) → (inl‘𝑢) ∈ ({∅} × 𝐴))
229, 21eqeltrrd 2894 . . . . . . . 8 ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢𝐴 ∧ (inl‘𝑢) = 𝑥)) → 𝑥 ∈ ({∅} × 𝐴))
238, 22rexlimddv 3253 . . . . . . 7 (𝑥 ∈ (inl “ 𝐴) → 𝑥 ∈ ({∅} × 𝐴))
24 elun1 4106 . . . . . . 7 (𝑥 ∈ ({∅} × 𝐴) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2523, 24syl 17 . . . . . 6 (𝑥 ∈ (inl “ 𝐴) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
26 df-dju 9318 . . . . . 6 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2725, 26eleqtrrdi 2904 . . . . 5 (𝑥 ∈ (inl “ 𝐴) → 𝑥 ∈ (𝐴𝐵))
28 djurf1o 9330 . . . . . . . . . . 11 inr:V–1-1-onto→({1o} × V)
29 f1ofn 6595 . . . . . . . . . . 11 (inr:V–1-1-onto→({1o} × V) → inr Fn V)
3028, 29ax-mp 5 . . . . . . . . . 10 inr Fn V
31 ssv 3942 . . . . . . . . . 10 𝐵 ⊆ V
32 fvelimab 6716 . . . . . . . . . 10 ((inr Fn V ∧ 𝐵 ⊆ V) → (𝑥 ∈ (inr “ 𝐵) ↔ ∃𝑢𝐵 (inr‘𝑢) = 𝑥))
3330, 31, 32mp2an 691 . . . . . . . . 9 (𝑥 ∈ (inr “ 𝐵) ↔ ∃𝑢𝐵 (inr‘𝑢) = 𝑥)
3433biimpi 219 . . . . . . . 8 (𝑥 ∈ (inr “ 𝐵) → ∃𝑢𝐵 (inr‘𝑢) = 𝑥)
35 simprr 772 . . . . . . . . 9 ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢𝐵 ∧ (inr‘𝑢) = 𝑥)) → (inr‘𝑢) = 𝑥)
36 opex 5324 . . . . . . . . . . 11 ⟨1o, 𝑢⟩ ∈ V
37 opeq2 4768 . . . . . . . . . . . 12 (𝑧 = 𝑢 → ⟨1o, 𝑧⟩ = ⟨1o, 𝑢⟩)
38 df-inr 9320 . . . . . . . . . . . 12 inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
3937, 38fvmptg 6747 . . . . . . . . . . 11 ((𝑢 ∈ V ∧ ⟨1o, 𝑢⟩ ∈ V) → (inr‘𝑢) = ⟨1o, 𝑢⟩)
4010, 36, 39mp2an 691 . . . . . . . . . 10 (inr‘𝑢) = ⟨1o, 𝑢
41 1oex 8097 . . . . . . . . . . . . 13 1o ∈ V
4241snid 4564 . . . . . . . . . . . 12 1o ∈ {1o}
43 opelxpi 5560 . . . . . . . . . . . 12 ((1o ∈ {1o} ∧ 𝑢𝐵) → ⟨1o, 𝑢⟩ ∈ ({1o} × 𝐵))
4442, 43mpan 689 . . . . . . . . . . 11 (𝑢𝐵 → ⟨1o, 𝑢⟩ ∈ ({1o} × 𝐵))
4544ad2antrl 727 . . . . . . . . . 10 ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢𝐵 ∧ (inr‘𝑢) = 𝑥)) → ⟨1o, 𝑢⟩ ∈ ({1o} × 𝐵))
4640, 45eqeltrid 2897 . . . . . . . . 9 ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢𝐵 ∧ (inr‘𝑢) = 𝑥)) → (inr‘𝑢) ∈ ({1o} × 𝐵))
4735, 46eqeltrrd 2894 . . . . . . . 8 ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢𝐵 ∧ (inr‘𝑢) = 𝑥)) → 𝑥 ∈ ({1o} × 𝐵))
4834, 47rexlimddv 3253 . . . . . . 7 (𝑥 ∈ (inr “ 𝐵) → 𝑥 ∈ ({1o} × 𝐵))
49 elun2 4107 . . . . . . 7 (𝑥 ∈ ({1o} × 𝐵) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
5048, 49syl 17 . . . . . 6 (𝑥 ∈ (inr “ 𝐵) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
5150, 26eleqtrrdi 2904 . . . . 5 (𝑥 ∈ (inr “ 𝐵) → 𝑥 ∈ (𝐴𝐵))
5227, 51jaoi 854 . . . 4 ((𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)) → 𝑥 ∈ (𝐴𝐵))
531, 52sylbi 220 . . 3 (𝑥 ∈ ((inl “ 𝐴) ∪ (inr “ 𝐵)) → 𝑥 ∈ (𝐴𝐵))
5453ssriv 3922 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐵)) ⊆ (𝐴𝐵)
55 djur 9336 . . . . 5 (𝑥 ∈ (𝐴𝐵) → (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
56 vex 3447 . . . . . . . . . 10 𝑦 ∈ V
57 f1odm 6598 . . . . . . . . . . 11 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
582, 57ax-mp 5 . . . . . . . . . 10 dom inl = V
5956, 58eleqtrri 2892 . . . . . . . . 9 𝑦 ∈ dom inl
60 simpl 486 . . . . . . . . 9 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑦𝐴)
6113funmpt2 6367 . . . . . . . . . 10 Fun inl
62 funfvima 6974 . . . . . . . . . 10 ((Fun inl ∧ 𝑦 ∈ dom inl) → (𝑦𝐴 → (inl‘𝑦) ∈ (inl “ 𝐴)))
6361, 62mpan 689 . . . . . . . . 9 (𝑦 ∈ dom inl → (𝑦𝐴 → (inl‘𝑦) ∈ (inl “ 𝐴)))
6459, 60, 63mpsyl 68 . . . . . . . 8 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) ∈ (inl “ 𝐴))
65 eleq1 2880 . . . . . . . . 9 (𝑥 = (inl‘𝑦) → (𝑥 ∈ (inl “ 𝐴) ↔ (inl‘𝑦) ∈ (inl “ 𝐴)))
6665adantl 485 . . . . . . . 8 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (𝑥 ∈ (inl “ 𝐴) ↔ (inl‘𝑦) ∈ (inl “ 𝐴)))
6764, 66mpbird 260 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ (inl “ 𝐴))
6867rexlimiva 3243 . . . . . 6 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ (inl “ 𝐴))
69 f1odm 6598 . . . . . . . . . . 11 (inr:V–1-1-onto→({1o} × V) → dom inr = V)
7028, 69ax-mp 5 . . . . . . . . . 10 dom inr = V
7156, 70eleqtrri 2892 . . . . . . . . 9 𝑦 ∈ dom inr
72 simpl 486 . . . . . . . . 9 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦𝐵)
73 f1ofun 6596 . . . . . . . . . . 11 (inr:V–1-1-onto→({1o} × V) → Fun inr)
7428, 73ax-mp 5 . . . . . . . . . 10 Fun inr
75 funfvima 6974 . . . . . . . . . 10 ((Fun inr ∧ 𝑦 ∈ dom inr) → (𝑦𝐵 → (inr‘𝑦) ∈ (inr “ 𝐵)))
7674, 75mpan 689 . . . . . . . . 9 (𝑦 ∈ dom inr → (𝑦𝐵 → (inr‘𝑦) ∈ (inr “ 𝐵)))
7771, 72, 76mpsyl 68 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) ∈ (inr “ 𝐵))
78 eleq1 2880 . . . . . . . . 9 (𝑥 = (inr‘𝑦) → (𝑥 ∈ (inr “ 𝐵) ↔ (inr‘𝑦) ∈ (inr “ 𝐵)))
7978adantl 485 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (𝑥 ∈ (inr “ 𝐵) ↔ (inr‘𝑦) ∈ (inr “ 𝐵)))
8077, 79mpbird 260 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ (inr “ 𝐵))
8180rexlimiva 3243 . . . . . 6 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ (inr “ 𝐵))
8268, 81orim12i 906 . . . . 5 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → (𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)))
8355, 82syl 17 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)))
8483, 1sylibr 237 . . 3 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ((inl “ 𝐴) ∪ (inr “ 𝐵)))
8584ssriv 3922 . 2 (𝐴𝐵) ⊆ ((inl “ 𝐴) ∪ (inr “ 𝐵))
8654, 85eqssi 3934 1 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wrex 3110  Vcvv 3444  cun 3882  wss 3884  c0 4246  {csn 4528  cop 4534   × cxp 5521  dom cdm 5523  cima 5526  Fun wfun 6322   Fn wfn 6323  1-1-ontowf1o 6327  cfv 6328  1oc1o 8082  cdju 9315  inlcinl 9316  inrcinr 9317
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-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-3or 1085  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-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  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-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  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-om 7565  df-1st 7675  df-2nd 7676  df-1o 8089  df-dju 9318  df-inl 9319  df-inr 9320
This theorem is referenced by: (None)
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