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Theorem dfres2 6041
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 6010 . 2 Rel (𝑅𝐴)
2 relopabv 5821 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 vex 3478 . . . 4 𝑧 ∈ V
4 vex 3478 . . . 4 𝑤 ∈ V
5 eleq1w 2816 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
6 breq1 5151 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
75, 6anbi12d 631 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
8 breq2 5152 . . . . 5 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
98anbi2d 629 . . . 4 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
103, 4, 7, 9opelopab 5542 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
114brresi 5990 . . 3 (𝑧(𝑅𝐴)𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 5149 . . 3 (𝑧(𝑅𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴))
1310, 11, 123bitr2ri 299 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
141, 2, 13eqrelriiv 5790 1 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  cop 4634   class class class wbr 5148  {copab 5210  cres 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-res 5688
This theorem is referenced by:  shftidt2  15032  bj-imdiridlem  36369  dfres4  37465  cnvepres  37470  ressn2  37615  tfsconcatrev  42400
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