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Theorem dfres2 6059
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 6023 . 2 Rel (𝑅𝐴)
2 relopabv 5831 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 vex 3484 . . . 4 𝑧 ∈ V
4 vex 3484 . . . 4 𝑤 ∈ V
5 eleq1w 2824 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
6 breq1 5146 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
75, 6anbi12d 632 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
8 breq2 5147 . . . . 5 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
98anbi2d 630 . . . 4 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
103, 4, 7, 9opelopab 5547 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
114brresi 6006 . . 3 (𝑧(𝑅𝐴)𝑤 ↔ (𝑧𝐴𝑧𝑅𝑤))
12 df-br 5144 . . 3 (𝑧(𝑅𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴))
1310, 11, 123bitr2ri 300 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
141, 2, 13eqrelriiv 5800 1 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143  {copab 5205  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  shftidt2  15120  bj-imdiridlem  37186  dfres4  38294  cnvepres  38299  ressn2  38443  tfsconcatrev  43361
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