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Theorem djulcl 9951
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djulcl (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))

Proof of Theorem djulcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inl 9943 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 4873 . . 3 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
3 elex 3500 . . 3 (𝐶𝐴𝐶 ∈ V)
4 0ex 5306 . . . . 5 ∅ ∈ V
54snid 4661 . . . 4 ∅ ∈ {∅}
6 opelxpi 5721 . . . 4 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
75, 6mpan 690 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
81, 2, 3, 7fvmptd3 7038 . 2 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
9 elun1 4181 . . . 4 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
107, 9syl 17 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
11 df-dju 9942 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1210, 11eleqtrrdi 2851 . 2 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
138, 12eqeltrd 2840 1 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3479  cun 3948  c0 4332  {csn 4625  cop 4631   × cxp 5682  cfv 6560  1oc1o 8500  cdju 9939  inlcinl 9940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-dju 9942  df-inl 9943
This theorem is referenced by:  inlresf  9955  updjudhcoinlf  9973
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