Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  djucllem GIF version

Theorem djucllem 11357
Description: Lemma for djulcl 6722 and djurcl 6723. (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 5313 . . 3 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 elex 2630 . . . 4 (𝐴𝐵𝐴 ∈ V)
3 djucllem.1 . . . . . 6 𝑋 ∈ V
43snid 3470 . . . . 5 𝑋 ∈ {𝑋}
5 opelxpi 4459 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝐴𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
64, 5mpan 415 . . . 4 (𝐴𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7 opeq2 3618 . . . . 5 (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8 djucllem.2 . . . . 5 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
97, 8fvmptg 5364 . . . 4 ((𝐴 ∈ V ∧ ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵)) → (𝐹𝐴) = ⟨𝑋, 𝐴⟩)
102, 6, 9syl2anc 403 . . 3 (𝐴𝐵 → (𝐹𝐴) = ⟨𝑋, 𝐴⟩)
111, 10eqtrd 2120 . 2 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
1211, 6eqeltrd 2164 1 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  {csn 3441  cop 3444  cmpt 3891   × cxp 4426  cres 4430  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-res 4440  df-iota 4967  df-fun 5004  df-fv 5010
This theorem is referenced by:  djulclALT  11358  djurclALT  11359
  Copyright terms: Public domain W3C validator