Theorem ressn 6177

Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
ressn	⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Proof of Theorem ressn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5909	. 2 ⊢ Rel (𝐴 ↾ {𝐵})
2		relxp 5598	. 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵}))
3		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3426	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	elimasn 5986	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
6		elsni 4575	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
7	6	sneqd 4570	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
8	7	imaeq2d 5958	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
9	8	eleq2d 2824	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
10	5, 9	bitr3id 284	. . . 4 ⊢ (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
11	10	pm5.32i 574	. . 3 ⊢ ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
12	4	opelresi 5888	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13		opelxp 5616	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
14	11, 12, 13	3bitr4i 302	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
15	1, 2, 14	eqrelriiv 5689	1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558  ⟨cop 4564   × cxp 5578   ↾ cres 5582   “ cima 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by:  gsum2dlem2  19487  dprd2da  19560  ustneism  23283