Theorem djucllem 15669

Description: Lemma for djulcl 7152 and djurcl 7153. (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 5599	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		elex 2782	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		djucllem.1	. . . . . 6 ⊢ 𝑋 ∈ V
4	3	snid 3663	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
5		opelxpi 4706	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝐴 ∈ 𝐵) → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
6	4, 5	mpan 424	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ⟨𝑋, 𝐴⟩ ∈ ({𝑋} × 𝐵))
7		opeq2 3819	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑋, 𝑥⟩ = ⟨𝑋, 𝐴⟩)
8		djucllem.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
9	7, 8	fvmptg 5654	. . . 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 2237	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ⟨𝑋, 𝐴⟩)
12	11, 6	eqeltrd 2281	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

Syntax hints:   → wi 4   = wceq 1372   ∈ wcel 2175  Vcvv 2771  {csn 3632  ⟨cop 3635   ↦ cmpt 4104   × cxp 4672   ↾ cres 4676  ‘cfv 5270

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-res 4686  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by:  djulclALT  15670  djurclALT  15671