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Theorem djurcl 6887
Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djurcl (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))

Proof of Theorem djurcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 2666 . . 3 (𝐶𝐵𝐶 ∈ V)
2 1oex 6273 . . . . 5 1o ∈ V
32snid 3520 . . . 4 1o ∈ {1o}
4 opelxpi 4529 . . . 4 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
53, 4mpan 418 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6 opeq2 3670 . . . 4 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7 df-inr 6883 . . . 4 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7fvmptg 5449 . . 3 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
91, 5, 8syl2anc 406 . 2 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10 elun2 3208 . . . 4 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
115, 10syl 14 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12 df-dju 6873 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1311, 12syl6eleqr 2206 . 2 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2189 1 (𝐶𝐵 → (inr‘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  cfv 5079  1oc1o 6258  cdju 6872  inrcinr 6881
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-13 1472  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  ax-un 4313
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-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-1o 6265  df-dju 6873  df-inr 6883
This theorem is referenced by:  updjudhcoinrg  6916  omp1eomlem  6929  difinfsnlem  6934  difinfsn  6935  0ct  6942  ctmlemr  6943  ctssdclemn0  6945  exmidfodomrlemr  7003  exmidfodomrlemrALT  7004
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