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Theorem djurclr 6901
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 5411 . 2 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶))
2 elex 2669 . . . 4 (𝐶𝐵𝐶 ∈ V)
3 1oex 6287 . . . . . 6 1o ∈ V
43snid 3524 . . . . 5 1o ∈ {1o}
5 opelxpi 4539 . . . . 5 ((1o ∈ {1o} ∧ 𝐶𝐵) → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
64, 5mpan 418 . . . 4 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵))
7 opeq2 3674 . . . . 5 (𝑥 = 𝐶 → ⟨1o, 𝑥⟩ = ⟨1o, 𝐶⟩)
8 df-inr 6899 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
97, 8fvmptg 5463 . . . 4 ((𝐶 ∈ V ∧ ⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵)) → (inr‘𝐶) = ⟨1o, 𝐶⟩)
102, 6, 9syl2anc 406 . . 3 (𝐶𝐵 → (inr‘𝐶) = ⟨1o, 𝐶⟩)
11 elun2 3212 . . . . 5 (⟨1o, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
126, 11syl 14 . . . 4 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
13 df-dju 6889 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1412, 13syl6eleqr 2209 . . 3 (𝐶𝐵 → ⟨1o, 𝐶⟩ ∈ (𝐴𝐵))
1510, 14eqeltrd 2192 . 2 (𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
161, 15eqeltrd 2192 1 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  Vcvv 2658  cun 3037  c0 3331  {csn 3495  cop 3498   × cxp 4505  cres 4509  cfv 5091  1oc1o 6272  cdju 6888  inrcinr 6897
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fv 5099  df-1o 6279  df-dju 6889  df-inr 6899
This theorem is referenced by:  inrresf1  6913
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