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Theorem djurclr 6984
 Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
Assertion
Ref Expression
djurclr (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))

Proof of Theorem djurclr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvres 5489 . 2 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶))
2 elex 2723 . . . 4 (𝐶𝐵𝐶 ∈ V)
3 1oex 6365 . . . . . 6 1o ∈ V
43snid 3591 . . . . 5 1o ∈ {1o}
5 opelxpi 4615 . . . . 5 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
64, 5mpan 421 . . . 4 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
7 opeq2 3742 . . . . 5 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
8 df-inr 6982 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
97, 8fvmptg 5541 . . . 4 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
102, 6, 9syl2anc 409 . . 3 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
11 elun2 3275 . . . . 5 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
126, 11syl 14 . . . 4 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13 df-dju 6972 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1412, 13eleqtrrdi 2251 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
1510, 14eqeltrd 2234 . 2 (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
161, 15eqeltrd 2234 1 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  Vcvv 2712   ∪ cun 3100  ∅c0 3394  {csn 3560  ⟨cop 3563   × cxp 4581   ↾ cres 4585  ‘cfv 5167  1oc1o 6350   ⊔ cdju 6971  inrcinr 6980 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-res 4595  df-iota 5132  df-fun 5169  df-fv 5175  df-1o 6357  df-dju 6972  df-inr 6982 This theorem is referenced by:  inrresf1  6996
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