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Theorem ressn 4801
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 4582 . 2 Rel (A ↾ {B})
2 relxp 4390 . 2 Rel ({B} × (A “ {B}))
3 ancom 253 . . . 4 ((⟨x, y A x {B}) ↔ (x {B} x, y A))
4 vex 2554 . . . . . . 7 x V
5 vex 2554 . . . . . . 7 y V
64, 5elimasn 4635 . . . . . 6 (y (A “ {x}) ↔ ⟨x, y A)
7 elsni 3391 . . . . . . . . 9 (x {B} → x = B)
87sneqd 3380 . . . . . . . 8 (x {B} → {x} = {B})
98imaeq2d 4611 . . . . . . 7 (x {B} → (A “ {x}) = (A “ {B}))
109eleq2d 2104 . . . . . 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 A x {B}) ↔ (x {B} y (A “ {B})))
145opelres 4560 . . 3 (⟨x, y (A ↾ {B}) ↔ (⟨x, y A x {B}))
15 opelxp 4317 . . 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 4377 1 (A ↾ {B}) = ({B} × (A “ {B}))
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {csn 3367  cop 3370   × cxp 4286  cres 4290  cima 4291
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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