ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  djulcl GIF version

Theorem djulcl 7044
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 2748 . . 3 (𝐶𝐴𝐶 ∈ V)
2 0ex 4127 . . . . 5 ∅ ∈ V
32snid 3622 . . . 4 ∅ ∈ {∅}
4 opelxpi 4655 . . . 4 ((∅ ∈ {∅} ∧ 𝐶𝐴) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
53, 4mpan 424 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
6 opeq2 3777 . . . 4 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
7 df-inl 7040 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
86, 7fvmptg 5588 . . 3 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
91, 5, 8syl2anc 411 . 2 (𝐶𝐴 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
10 elun1 3302 . . . 4 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
115, 10syl 14 . . 3 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12 df-dju 7031 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1311, 12eleqtrrdi 2271 . 2 (𝐶𝐴 → ⟨∅, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2254 1 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127  c0 3422  {csn 3591  cop 3594   × cxp 4621  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-dju 7031  df-inl 7040
This theorem is referenced by:  djulclb  7048  updjudhcoinlf  7073  omp1eomlem  7087  difinfsnlem  7092  difinfsn  7093  ctmlemr  7101  ctm  7102  ctssdclemn0  7103  ctssdccl  7104  fodju0  7139  exmidfodomrlemr  7195  exmidfodomrlemrALT  7196  subctctexmid  14399
  Copyright terms: Public domain W3C validator