Theorem djucllem 16828

Description: Lemma for djulcl 7391 and djurcl 7392. (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 5719	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		elex 2833	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		djucllem.1	. . . . . 6 ⊢ 𝑋 ∈ V
4	3	snid 3740	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
5		opelxpi 4806	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
6	4, 5	mpan 428	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7		opeq2 3905	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8		djucllem.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
9	7, 8	fvmptg 5781	. . . 4 ⊢ ((𝐴 ∈ V ∧ ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵)) → (𝐹‘𝐴) = ⟨𝑋, 𝐴⟩)
10	2, 6, 9	syl2anc 415	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹‘𝐴) = ⟨𝑋, 𝐴⟩)
11	1, 10	eqtrd 2271	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
12	11, 6	eqeltrd 2315	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3709  ⟨cop 3712   ↦ cmpt 4192   × cxp 4772   ↾ cres 4776  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385

This theorem is used by:  djulclALT  16829  djurclALT  16830