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Theorem ressn 5074
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.
StepHypRef Expression
1 relres 4842 . 2 Rel (𝐴 ↾ {𝐵})
2 relxp 4643 . 2 Rel ({𝐵} × (𝐴 “ {𝐵}))
3 ancom 264 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4 vex 2684 . . . . . . 7 𝑥 ∈ V
5 vex 2684 . . . . . . 7 𝑦 ∈ V
64, 5elimasn 4901 . . . . . 6 (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7 elsni 3540 . . . . . . . . 9 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
87sneqd 3535 . . . . . . . 8 (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
98imaeq2d 4876 . . . . . . 7 (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
109eleq2d 2207 . . . . . 6 (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
116, 10syl5bbr 193 . . . . 5 (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ (𝐴 “ {𝐵})))
1211pm5.32i 449 . . . 4 ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
133, 12bitri 183 . . 3 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
145opelres 4819 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}))
15 opelxp 4564 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
1613, 14, 153bitr4i 211 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
171, 2, 16eqrelriiv 4628 1 (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1331  wcel 1480  {csn 3522  cop 3525   × cxp 4532  cres 4536  cima 4537
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547
This theorem is referenced by: (None)
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