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Theorem djurcl 9316
 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 3489 . . 3 (𝐶𝐵𝐶 ∈ V)
2 1oex 8085 . . . . 5 1o ∈ V
32snid 4574 . . . 4 1o ∈ {1o}
4 opelxpi 5565 . . . 4 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
53, 4mpan 689 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
6 opeq2 4777 . . . 4 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
7 df-inr 9308 . . . 4 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
86, 7fvmptg 6739 . . 3 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
91, 5, 8syl2anc 587 . 2 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
10 elun2 4129 . . . 4 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
115, 10syl 17 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
12 df-dju 9306 . . 3 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1311, 12eleqtrrdi 2923 . 2 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
149, 13eqeltrd 2912 1 (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3471   ∪ cun 3908  ∅c0 4266  {csn 4540  ⟨cop 4546   × cxp 5526  ‘cfv 6328  1oc1o 8070   ⊔ cdju 9303  inrcinr 9305 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-ord 6167  df-on 6168  df-suc 6170  df-iota 6287  df-fun 6330  df-fv 6336  df-1o 8077  df-dju 9306  df-inr 9308 This theorem is referenced by:  inrresf  9321  updjudhcoinrg  9338
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