Theorem djucllem 16396

Description: Lemma for djulcl 7249 and djurcl 7250. (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 5663	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		elex 2814	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		djucllem.1	. . . . . 6 ⊢ 𝑋 ∈ V
4	3	snid 3700	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
5		opelxpi 4757	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7		opeq2 3863	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8		djucllem.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
9	7, 8	fvmptg 5722	. . . 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 2264	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
12	11, 6	eqeltrd 2308	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802  {csn 3669  ⟨cop 3672   ↦ cmpt 4150   × cxp 4723   ↾ cres 4727  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by:  djulclALT  16397  djurclALT  16398