Theorem djucllem 16572

Description: Lemma for djulcl 7342 and djurcl 7343. (Contributed by BJ, 4-Jul-2022.)

Hypotheses

Ref	Expression
djucllem.1	⊢ 𝑋 ∈ V
djucllem.2	⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)

Assertion

Ref	Expression
djucllem	⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem djucllem

Step	Hyp	Ref	Expression
1		fvres 5694	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		elex 2825	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		djucllem.1	. . . . . 6 ⊢ 𝑋 ∈ V
4	3	snid 3720	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
5		opelxpi 4781	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7		opeq2 3884	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8		djucllem.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
9	7, 8	fvmptg 5753	. . . 4 ⊢ ((𝐴 ∈ V ∧ ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵)) → (𝐹‘𝐴) = ⟨𝑋, 𝐴⟩)
10	2, 6, 9	syl2anc 411	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹‘𝐴) = ⟨𝑋, 𝐴⟩)
11	1, 10	eqtrd 2265	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
12	11, 6	eqeltrd 2309	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813  {csn 3689  ⟨cop 3692   ↦ cmpt 4171   × cxp 4747   ↾ cres 4751  ‘cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by:  djulclALT  16573  djurclALT  16574