Theorem ressn 5323

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 5086	. 2 ⊢ Rel (𝐴 ↾ {𝐵})
2		relxp 4879	. 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵}))
3		ancom 266	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 2824	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	elimasn 5149	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7		elsni 3723	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
8	7	sneqd 3718	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
9	8	imaeq2d 5121	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
10	9	eleq2d 2308	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
11	6, 10	bitr3id 194	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
12	11	pm5.32i 458	. . . 4 ⊢ ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
13	3, 12	bitri 184	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
14	5	opelres 5063	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}))
15		opelxp 4799	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
16	13, 14, 15	3bitr4i 212	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
17	1, 2, 16	eqrelriiv 4864	1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Syntax hints:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {csn 3705  ⟨cop 3708   × cxp 4767   ↾ cres 4771   “ cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by: (None)