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Theorem ressn 4971
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 4741 . 2 Rel (𝐴 ↾ {𝐵})
2 relxp 4547 . 2 Rel ({𝐵} × (𝐴 “ {𝐵}))
3 ancom 262 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4 vex 2622 . . . . . . 7 𝑥 ∈ V
5 vex 2622 . . . . . . 7 𝑦 ∈ V
64, 5elimasn 4799 . . . . . 6 (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7 elsni 3464 . . . . . . . . 9 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
87sneqd 3459 . . . . . . . 8 (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
98imaeq2d 4774 . . . . . . 7 (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
109eleq2d 2157 . . . . . 6 (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
116, 10syl5bbr 192 . . . . 5 (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ (𝐴 “ {𝐵})))
1211pm5.32i 442 . . . 4 ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
133, 12bitri 182 . . 3 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
145opelres 4718 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ {𝐵}))
15 opelxp 4467 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
1613, 14, 153bitr4i 210 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
171, 2, 16eqrelriiv 4532 1 (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  {csn 3446  cop 3449   × cxp 4436  cres 4440  cima 4441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by: (None)
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