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Theorem djucllem 14201
Description: Lemma for djulcl 7044 and djurcl 7045. (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 5535 . . 3 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 elex 2748 . . . 4 (𝐴𝐵𝐴 ∈ V)
3 djucllem.1 . . . . . 6 𝑋 ∈ V
43snid 3622 . . . . 5 𝑋 ∈ {𝑋}
5 opelxpi 4655 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝐴𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
64, 5mpan 424 . . . 4 (𝐴𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7 opeq2 3777 . . . . 5 (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8 djucllem.2 . . . . 5 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
97, 8fvmptg 5588 . . . 4 ((𝐴 ∈ V ∧ ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵)) → (𝐹𝐴) = ⟨𝑋, 𝐴⟩)
102, 6, 9syl2anc 411 . . 3 (𝐴𝐵 → (𝐹𝐴) = ⟨𝑋, 𝐴⟩)
111, 10eqtrd 2210 . 2 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
1211, 6eqeltrd 2254 1 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  {csn 3591  cop 3594  cmpt 4061   × cxp 4621  cres 4625  cfv 5212
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-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-pow 4171  ax-pr 4206
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-un 3133  df-in 3135  df-ss 3142  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 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-res 4635  df-iota 5174  df-fun 5214  df-fv 5220
This theorem is referenced by:  djulclALT  14202  djurclALT  14203
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