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Theorem djurcl 7053
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 2750 . . 3 (𝐶𝐵𝐶 ∈ V)
2 1oex 6427 . . . . 5 1o ∈ V
32snid 3625 . . . 4 1o ∈ {1o}
4 opelxpi 4660 . . . 4 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
53, 4mpan 424 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6 opeq2 3781 . . . 4 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7 df-inr 7049 . . . 4 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7fvmptg 5594 . . 3 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
91, 5, 8syl2anc 411 . 2 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10 elun2 3305 . . . 4 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
115, 10syl 14 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12 df-dju 7039 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1311, 12eleqtrrdi 2271 . 2 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2254 1 (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2739  cun 3129  c0 3424  {csn 3594  cop 3597   × cxp 4626  cfv 5218  1oc1o 6412  cdju 7038  inrcinr 7047
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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-1o 6419  df-dju 7039  df-inr 7049
This theorem is referenced by:  updjudhcoinrg  7082  omp1eomlem  7095  difinfsnlem  7100  difinfsn  7101  0ct  7108  ctmlemr  7109  ctssdclemn0  7111  exmidfodomrlemr  7203  exmidfodomrlemrALT  7204
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