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Theorem dfres2 5874
 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 5845 . 2 Rel (𝑅𝐴)
2 relopab 5658 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 vex 3411 . . . 4 𝑧 ∈ V
4 vex 3411 . . . 4 𝑤 ∈ V
5 eleq1w 2833 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
6 breq1 5028 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
75, 6anbi12d 634 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
8 breq2 5029 . . . . 5 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
98anbi2d 632 . . . 4 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
103, 4, 7, 9opelopab 5392 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
114brresi 5825 . . 3 (𝑧(𝑅𝐴)𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 5026 . . 3 (𝑧(𝑅𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴))
1310, 11, 123bitr2ri 304 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
141, 2, 13eqrelriiv 5625 1 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ⟨cop 4521   class class class wbr 5025  {copab 5087   ↾ cres 5519 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-xp 5523  df-rel 5524  df-res 5529 This theorem is referenced by:  shftidt2  14473  bj-imdiridlem  34865  dfres4  35975  cnvepres  35980
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