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Theorem djulcl 9934
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 9926 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 4875 . . 3 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
3 elex 3490 . . 3 (𝐶𝐴𝐶 ∈ V)
4 0ex 5307 . . . . 5 ∅ ∈ V
54snid 4665 . . . 4 ∅ ∈ {∅}
6 opelxpi 5715 . . . 4 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
75, 6mpan 689 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
81, 2, 3, 7fvmptd3 7028 . 2 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
9 elun1 4176 . . . 4 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
107, 9syl 17 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
11 df-dju 9925 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1210, 11eleqtrrdi 2840 . 2 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
138, 12eqeltrd 2829 1 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3471  cun 3945  c0 4323  {csn 4629  cop 4635   × cxp 5676  cfv 6548  1oc1o 8480  cdju 9922  inlcinl 9923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-dju 9925  df-inl 9926
This theorem is referenced by:  inlresf  9938  updjudhcoinlf  9956
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