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Theorem ressn 6123
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 5869 . 2 Rel (𝐴 ↾ {𝐵})
2 relxp 5560 . 2 Rel ({𝐵} × (𝐴 “ {𝐵}))
3 vex 3483 . . . . . 6 𝑥 ∈ V
4 vex 3483 . . . . . 6 𝑦 ∈ V
53, 4elimasn 5941 . . . . 5 (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
6 elsni 4567 . . . . . . . 8 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
76sneqd 4562 . . . . . . 7 (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
87imaeq2d 5916 . . . . . 6 (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
98eleq2d 2901 . . . . 5 (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
105, 9bitr3id 288 . . . 4 (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ (𝐴 “ {𝐵})))
1110pm5.32i 578 . . 3 ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
124opelresi 5848 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13 opelxp 5578 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
1411, 12, 133bitr4i 306 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
151, 2, 14eqrelriiv 5650 1 (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  {csn 4550  cop 4556   × cxp 5540  cres 5544  cima 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  gsum2dlem2  19091  dprd2da  19164  ustneism  22836
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