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Theorem dmres 4575
Description: The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
dmres dom (AB) = (B ∩ dom A)

Proof of Theorem dmres
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 x V
21eldm2 4476 . . . 4 (x dom (AB) ↔ yx, y (AB))
3 19.41v 1779 . . . . 5 (y(⟨x, y A x B) ↔ (yx, y A x B))
4 vex 2554 . . . . . . 7 y V
54opelres 4560 . . . . . 6 (⟨x, y (AB) ↔ (⟨x, y A x B))
65exbii 1493 . . . . 5 (yx, y (AB) ↔ y(⟨x, y A x B))
71eldm2 4476 . . . . . 6 (x dom Ayx, y A)
87anbi1i 431 . . . . 5 ((x dom A x B) ↔ (yx, y A x B))
93, 6, 83bitr4i 201 . . . 4 (yx, y (AB) ↔ (x dom A x B))
102, 9bitr2i 174 . . 3 ((x dom A x B) ↔ x dom (AB))
1110ineqri 3124 . 2 (dom AB) = dom (AB)
12 incom 3123 . 2 (dom AB) = (B ∩ dom A)
1311, 12eqtr3i 2059 1 dom (AB) = (B ∩ dom A)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  cin 2910  cop 3370  dom cdm 4288  cres 4290
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  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-dm 4298  df-res 4300
This theorem is referenced by:  ssdmres  4576  dmresexg  4577  imadisj  4630  ndmima  4645  imainrect  4709  dmresv  4722  resdmres  4755  funimacnv  4918  fnresdisj  4952  fnres  4958  ssimaex  5177  fnreseql  5220  respreima  5238  ffvresb  5271  fsnunfv  5306  funfvima  5333  offres  5704  smores  5848  smores3  5849  smores2  5850  dmaddpi  6309  dmmulpi  6310
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