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Theorem djulcl 6722
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 elex 2630 . . 3 (𝐶𝐴𝐶 ∈ V)
2 0ex 3958 . . . . 5 ∅ ∈ V
32snid 3470 . . . 4 ∅ ∈ {∅}
4 opelxpi 4459 . . . 4 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
53, 4mpan 415 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
6 opeq2 3618 . . . 4 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
7 df-inl 6718 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
86, 7fvmptg 5364 . . 3 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
91, 5, 8syl2anc 403 . 2 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
10 elun1 3165 . . . 4 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵)))
115, 10syl 14 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵)))
12 df-dju 6710 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵))
1311, 12syl6eleqr 2181 . 2 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2164 1 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  cun 2995  c0 3284  {csn 3441  cop 3444   × cxp 4426  cfv 5002  1𝑜c1o 6156  cdju 6709  inlcinl 6716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-dju 6710  df-inl 6718
This theorem is referenced by:  djulclb  6726  updjudhcoinlf  6750  fodjuomnilem0  6781  exmidfodomrlemr  6807  exmidfodomrlemrALT  6808
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