Theorem ressn 5150

Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

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 4918	. 2 ⊢ Rel (𝐴 ↾ {𝐵})
2		relxp 4719	. 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵}))
3		ancom 264	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4		vex 2733	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 2733	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	elimasn 4977	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7		elsni 3600	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
8	7	sneqd 3595	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
9	8	imaeq2d 4952	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
10	9	eleq2d 2240	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
11	6, 10	bitr3id 193	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
12	11	pm5.32i 451	. . . 4 ⊢ ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
13	3, 12	bitri 183	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
14	5	opelres 4895	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}))
15		opelxp 4640	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
16	13, 14, 15	3bitr4i 211	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
17	1, 2, 16	eqrelriiv 4704	1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {csn 3582  ⟨cop 3585   × cxp 4608   ↾ cres 4612   “ cima 4613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4106  ax-pow 4159  ax-pr 4193

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3567  df-sn 3588  df-pr 3589  df-op 3591  df-br 3989  df-opab 4050  df-xp 4616  df-rel 4617  df-cnv 4618  df-dm 4620  df-rn 4621  df-res 4622  df-ima 4623

This theorem is referenced by: (None)