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Theorem dfres2 4959
Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
Assertion
Ref Expression
dfres2 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem dfres2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4935 . 2 Rel (𝑅𝐴)
2 relopab 4753 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 vex 2740 . . . . 5 𝑤 ∈ V
43brres 4913 . . . 4 (𝑧(𝑅𝐴)𝑤 ↔ (𝑧𝑅𝑤𝑧𝐴))
5 df-br 4004 . . . 4 (𝑧(𝑅𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴))
6 ancom 266 . . . 4 ((𝑧𝑅𝑤𝑧𝐴) ↔ (𝑧𝐴𝑧𝑅𝑤))
74, 5, 63bitr3i 210 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ (𝑧𝐴𝑧𝑅𝑤))
8 vex 2740 . . . 4 𝑧 ∈ V
9 eleq1 2240 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
10 breq1 4006 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
119, 10anbi12d 473 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
12 breq2 4007 . . . . 5 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
1312anbi2d 464 . . . 4 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
148, 3, 11, 13opelopab 4271 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
157, 14bitr4i 187 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
161, 2, 15eqrelriiv 4720 1 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  cop 3595   class class class wbr 4003  {copab 4063  cres 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-res 4638
This theorem is referenced by:  shftidt2  10836
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