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Theorem djurcl 9954
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 3482 . . 3 (𝐶𝐵𝐶 ∈ V)
2 1oex 8506 . . . . 5 1o ∈ V
32snid 4669 . . . 4 1o ∈ {1o}
4 opelxpi 5719 . . . 4 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
53, 4mpan 688 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6 opeq2 4880 . . . 4 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7 df-inr 9946 . . . 4 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7fvmptg 7007 . . 3 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
91, 5, 8syl2anc 582 . 2 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10 elun2 4178 . . . 4 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
115, 10syl 17 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12 df-dju 9944 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1311, 12eleqtrrdi 2837 . 2 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2826 1 (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3462  cun 3945  c0 4325  {csn 4633  cop 4639   × cxp 5680  cfv 6554  1oc1o 8489  cdju 9941  inrcinr 9943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-suc 6382  df-iota 6506  df-fun 6556  df-fv 6562  df-1o 8496  df-dju 9944  df-inr 9946
This theorem is referenced by:  inrresf  9959  updjudhcoinrg  9976
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