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Theorem djurclr 7048
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 5539 . 2 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶))
2 elex 2748 . . . 4 (𝐶𝐵𝐶 ∈ V)
3 1oex 6424 . . . . . 6 1o ∈ V
43snid 3623 . . . . 5 1o ∈ {1o}
5 opelxpi 4658 . . . . 5 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
64, 5mpan 424 . . . 4 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
7 opeq2 3779 . . . . 5 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
8 df-inr 7046 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
97, 8fvmptg 5592 . . . 4 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
102, 6, 9syl2anc 411 . . 3 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
11 elun2 3303 . . . . 5 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
126, 11syl 14 . . . 4 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13 df-dju 7036 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1412, 13eleqtrrdi 2271 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
1510, 14eqeltrd 2254 . 2 (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
161, 15eqeltrd 2254 1 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127  c0 3422  {csn 3592  cop 3595   × cxp 4624  cres 4628  cfv 5216  1oc1o 6409  cdju 7035  inrcinr 7044
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fv 5224  df-1o 6416  df-dju 7036  df-inr 7046
This theorem is referenced by:  inrresf1  7060
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