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Theorem dmres 4654
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 2605 . . . . 5 𝑥 ∈ V
21eldm2 4555 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
3 19.41v 1824 . . . . 5 (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥𝐵))
4 vex 2605 . . . . . . 7 𝑦 ∈ V
54opelres 4639 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥𝐵))
65exbii 1537 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥𝐵))
71eldm2 4555 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
87anbi1i 446 . . . . 5 ((𝑥 ∈ dom 𝐴𝑥𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥𝐵))
93, 6, 83bitr4i 210 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥𝐵))
102, 9bitr2i 183 . . 3 ((𝑥 ∈ dom 𝐴𝑥𝐵) ↔ 𝑥 ∈ dom (𝐴𝐵))
1110ineqri 3160 . 2 (dom 𝐴𝐵) = dom (𝐴𝐵)
12 incom 3159 . 2 (dom 𝐴𝐵) = (𝐵 ∩ dom 𝐴)
1311, 12eqtr3i 2104 1 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  cin 2973  cop 3403  dom cdm 4365  cres 4367
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-dm 4375  df-res 4377
This theorem is referenced by:  ssdmres  4655  dmresexg  4656  imadisj  4711  ndmima  4726  imainrect  4790  dmresv  4803  resdmres  4836  funimacnv  5000  fnresdisj  5034  fnres  5040  ssimaex  5260  fnreseql  5303  respreima  5321  ffvresb  5354  fsnunfv  5389  funfvima  5416  offres  5787  smores  5935  smores3  5936  smores2  5937  fnfi  6436  dmaddpi  6566  dmmulpi  6567
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