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Theorem ressn 5913
 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 5663 . 2 Rel (𝐴 ↾ {𝐵})
2 relxp 5361 . 2 Rel ({𝐵} × (𝐴 “ {𝐵}))
3 vex 3418 . . . . . 6 𝑥 ∈ V
4 vex 3418 . . . . . 6 𝑦 ∈ V
53, 4elimasn 5732 . . . . 5 (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
6 elsni 4415 . . . . . . . 8 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
76sneqd 4410 . . . . . . 7 (𝑥 ∈ {𝐵} → {𝑥} = {𝐵})
87imaeq2d 5708 . . . . . 6 (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵}))
98eleq2d 2893 . . . . 5 (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵})))
105, 9syl5bbr 277 . . . 4 (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ (𝐴 “ {𝐵})))
1110pm5.32i 572 . . 3 ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
124opelresi 5638 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
13 opelxp 5379 . . 3 (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵})))
1411, 12, 133bitr4i 295 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})))
151, 2, 14eqrelriiv 5449 1 (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {csn 4398  ⟨cop 4404   × cxp 5341   ↾ cres 5345   “ cima 5346 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356 This theorem is referenced by:  gsum2dlem2  18724  dprd2da  18796  ustneism  22398
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