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Theorem ressn 4858
 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 4639 . 2 Rel (𝐴 ↾ {𝐵})
2 relxp 4447 . 2 Rel ({𝐵} × (𝐴 “ {𝐵}))
3 ancom 253 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4 vex 2560 . . . . . . 7 𝑥 ∈ V
5 vex 2560 . . . . . . 7 𝑦 ∈ V
64, 5elimasn 4692 . . . . . 6 (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7 elsni 3393 . . . . . . . . 9 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
87sneqd 3388 . . . . . . . 8 (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
98imaeq2d 4668 . . . . . . 7 (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
109eleq2d 2107 . . . . . 6 (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
116, 10syl5bbr 183 . . . . 5 (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ (𝐴 “ {𝐵})))
1211pm5.32i 427 . . . 4 ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
133, 12bitri 173 . . 3 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
145opelres 4617 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}))
15 opelxp 4374 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
1613, 14, 153bitr4i 201 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
171, 2, 16eqrelriiv 4434 1 (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {csn 3375  ⟨cop 3378   × cxp 4343   ↾ cres 4347   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by: (None)
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