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Theorem djulclALT 13172
Description: Shortening of djulcl 6942 using djucllem 13171. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
djulclALT (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))

Proof of Theorem djulclALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0ex 4061 . . . . 5 ∅ ∈ V
2 df-inl 6938 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
31, 2djucllem 13171 . . . 4 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴))
43orcd 723 . . 3 (𝐶𝐴 → (((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inl ↾ 𝐴)‘𝐶) ∈ ({1o} × 𝐵)))
5 elun 3220 . . 3 (((inl ↾ 𝐴)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inl ↾ 𝐴)‘𝐶) ∈ ({1o} × 𝐵)))
64, 5sylibr 133 . 2 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
7 df-dju 6929 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
86, 7eleqtrrdi 2234 1 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 698  wcel 1481  cun 3072  c0 3366  {csn 3530   × cxp 4543  cres 4547  cfv 5129  1oc1o 6312  cdju 6928  inlcinl 6936
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-res 4557  df-iota 5094  df-fun 5131  df-fv 5137  df-dju 6929  df-inl 6938
This theorem is referenced by: (None)
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