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Theorem djurclALT 12590
Description: Shortening of djurcl 6852 using djucllem 12588. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
djurclALT (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))

Proof of Theorem djurclALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1oex 6251 . . . . 5 1o ∈ V
2 df-inr 6848 . . . . 5 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
31, 2djucllem 12588 . . . 4 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ ({1o} × 𝐵))
43olcd 694 . . 3 (𝐶𝐵 → (((inr ↾ 𝐵)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inr ↾ 𝐵)‘𝐶) ∈ ({1o} × 𝐵)))
5 elun 3164 . . 3 (((inr ↾ 𝐵)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (((inr ↾ 𝐵)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inr ↾ 𝐵)‘𝐶) ∈ ({1o} × 𝐵)))
64, 5sylibr 133 . 2 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
7 df-dju 6838 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
86, 7syl6eleqr 2193 1 (𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 670  wcel 1448  cun 3019  c0 3310  {csn 3474   × cxp 4475  cres 4479  cfv 5059  1oc1o 6236  cdju 6837  inrcinr 6846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-res 4489  df-iota 5024  df-fun 5061  df-fv 5067  df-1o 6243  df-dju 6838  df-inr 6848
This theorem is referenced by: (None)
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