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Theorem djulclr 6884
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
Assertion
Ref Expression
djulclr (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))

Proof of Theorem djulclr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvres 5397 . 2 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶))
2 elex 2666 . . . 4 (𝐶𝐴𝐶 ∈ V)
3 0ex 4013 . . . . . 6 ∅ ∈ V
43snid 3520 . . . . 5 ∅ ∈ {∅}
5 opelxpi 4529 . . . . 5 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
64, 5mpan 418 . . . 4 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
7 opeq2 3670 . . . . 5 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
8 df-inl 6882 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
97, 8fvmptg 5449 . . . 4 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
102, 6, 9syl2anc 406 . . 3 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
11 elun1 3207 . . . . 5 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
126, 11syl 14 . . . 4 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13 df-dju 6873 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1412, 13syl6eleqr 2206 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
1510, 14eqeltrd 2189 . 2 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
161, 15eqeltrd 2189 1 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1312  wcel 1461  Vcvv 2655  cun 3033  c0 3327  {csn 3491  cop 3494   × cxp 4495  cres 4499  cfv 5079  1oc1o 6258  cdju 6872  inlcinl 6880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-res 4509  df-iota 5044  df-fun 5081  df-fv 5087  df-dju 6873  df-inl 6882
This theorem is referenced by:  inlresf1  6896
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