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Theorem djulclALT 14175
Description: Shortening of djulcl 7043 using djucllem 14174. (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 4127 . . . . 5 ∅ ∈ V
2 df-inl 7039 . . . . 5 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
31, 2djucllem 14174 . . . 4 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴))
43orcd 733 . . 3 (𝐶𝐴 → (((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inl ↾ 𝐴)‘𝐶) ∈ ({1o} × 𝐵)))
5 elun 3276 . . 3 (((inl ↾ 𝐴)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inl ↾ 𝐴)‘𝐶) ∈ ({1o} × 𝐵)))
64, 5sylibr 134 . 2 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
7 df-dju 7030 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
86, 7eleqtrrdi 2271 1 (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708  wcel 2148  cun 3127  c0 3422  {csn 3591   × cxp 4620  cres 4624  cfv 5211  1oc1o 6403  cdju 7029  inlcinl 7037
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-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-dju 7030  df-inl 7039
This theorem is referenced by: (None)
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