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Theorem djulcl 6902
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 2669 . . 3 (𝐶𝐴𝐶 ∈ V)
2 0ex 4023 . . . . 5 ∅ ∈ V
32snid 3524 . . . 4 ∅ ∈ {∅}
4 opelxpi 4539 . . . 4 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
53, 4mpan 418 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
6 opeq2 3674 . . . 4 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
7 df-inl 6898 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
86, 7fvmptg 5463 . . 3 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
91, 5, 8syl2anc 406 . 2 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
10 elun1 3211 . . . 4 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
115, 10syl 14 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12 df-dju 6889 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1311, 12syl6eleqr 2209 . 2 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2192 1 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  Vcvv 2658  cun 3037  c0 3331  {csn 3495  cop 3498   × cxp 4505  cfv 5091  1oc1o 6272  cdju 6888  inlcinl 6896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-dju 6889  df-inl 6898
This theorem is referenced by:  djulclb  6906  updjudhcoinlf  6931  omp1eomlem  6945  difinfsnlem  6950  difinfsn  6951  ctmlemr  6959  ctm  6960  ctssdclemn0  6961  ctssdccl  6962  fodju0  6985  exmidfodomrlemr  7022  exmidfodomrlemrALT  7023  subctctexmid  13030
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