Theorem ressn 6283

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 5998	. 2 ⊢ Rel (𝐴 ↾ {𝐵})
2		relxp 5673	. 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵}))
3		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	elimasn 6086	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
6		elsni 4601	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
7	6	sneqd 4596	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
8	7	imaeq2d 6056	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
9	8	eleq2d 2846	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
10	5, 9	bitr3id 288	. . . 4 ⊢ (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
11	10	pm5.32i 585	. . 3 ⊢ ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
12	4	opelresi 5980	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13		opelxp 5691	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
14	11, 12, 13	3bitr4i 306	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
15	1, 2, 14	eqrelriiv 5770	1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4584  ⟨cop 4590   × cxp 5653   ↾ cres 5657   “ cima 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  gsum2dlem2  20099  dprd2da  20172  ustneism  24451  tposres3  49808