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Theorem djucllem 13091
Description: Lemma for djulcl 6936 and djurcl 6937. (Contributed by BJ, 4-Jul-2022.)
Hypotheses
Ref Expression
djucllem.1 𝑋 ∈ V
djucllem.2 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
Assertion
Ref Expression
djucllem (𝐴𝐵 → ((𝐹𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem djucllem
StepHypRef Expression
1 fvres 5445 . . 3 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 elex 2697 . . . 4 (𝐴𝐵𝐴 ∈ V)
3 djucllem.1 . . . . . 6 𝑋 ∈ V
43snid 3556 . . . . 5 𝑋 ∈ {𝑋}
5 opelxpi 4571 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝐴𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
64, 5mpan 420 . . . 4 (𝐴𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7 opeq2 3706 . . . . 5 (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8 djucllem.2 . . . . 5 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
97, 8fvmptg 5497 . . . 4 ((𝐴 ∈ V ∧ ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵)) → (𝐹𝐴) = ⟨𝑋, 𝐴⟩)
102, 6, 9syl2anc 408 . . 3 (𝐴𝐵 → (𝐹𝐴) = ⟨𝑋, 𝐴⟩)
111, 10eqtrd 2172 . 2 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
1211, 6eqeltrd 2216 1 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2686  {csn 3527  cop 3530  cmpt 3989   × cxp 4537  cres 4541  cfv 5123
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131
This theorem is referenced by:  djulclALT  13092  djurclALT  13093
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