Theorem ressn 6232

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 5953	. 2 ⊢ Rel (𝐴 ↾ {𝐵})
2		relxp 5632	. 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵}))
3		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3440	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	elimasn 6038	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
6		elsni 4590	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
7	6	sneqd 4585	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
8	7	imaeq2d 6008	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
9	8	eleq2d 2817	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
10	5, 9	bitr3id 285	. . . 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 5935	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13		opelxp 5650	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
14	11, 12, 13	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
15	1, 2, 14	eqrelriiv 5729	1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {csn 4573  ⟨cop 4579   × cxp 5612   ↾ cres 5616   “ cima 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627

This theorem is referenced by:  gsum2dlem2  19883  dprd2da  19956  ustneism  24139  tposres3  48991