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Theorem djulclr 7078
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 5558 . 2 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶))
2 elex 2763 . . . 4 (𝐶𝐴𝐶 ∈ V)
3 0ex 4145 . . . . . 6 ∅ ∈ V
43snid 3638 . . . . 5 ∅ ∈ {∅}
5 opelxpi 4676 . . . . 5 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
64, 5mpan 424 . . . 4 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
7 opeq2 3794 . . . . 5 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
8 df-inl 7076 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
97, 8fvmptg 5613 . . . 4 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
102, 6, 9syl2anc 411 . . 3 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
11 elun1 3317 . . . . 5 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
126, 11syl 14 . . . 4 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13 df-dju 7067 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1412, 13eleqtrrdi 2283 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
1510, 14eqeltrd 2266 . 2 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
161, 15eqeltrd 2266 1 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142  c0 3437  {csn 3607  cop 3610   × cxp 4642  cres 4646  cfv 5235  1oc1o 6434  cdju 7066  inlcinl 7074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-dju 7067  df-inl 7076
This theorem is referenced by:  inlresf1  7090
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