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Theorem dfres2 5412
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 5385 . 2 Rel (𝑅𝐴)
2 relopab 5207 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
3 vex 3189 . . . . 5 𝑤 ∈ V
43brres 5362 . . . 4 (𝑧(𝑅𝐴)𝑤 ↔ (𝑧𝑅𝑤𝑧𝐴))
5 df-br 4614 . . . 4 (𝑧(𝑅𝐴)𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴))
6 ancom 466 . . . 4 ((𝑧𝑅𝑤𝑧𝐴) ↔ (𝑧𝐴𝑧𝑅𝑤))
74, 5, 63bitr3i 290 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ (𝑧𝐴𝑧𝑅𝑤))
8 vex 3189 . . . 4 𝑧 ∈ V
9 eleq1 2686 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
10 breq1 4616 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
119, 10anbi12d 746 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑦)))
12 breq2 4617 . . . . 5 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
1312anbi2d 739 . . . 4 (𝑦 = 𝑤 → ((𝑧𝐴𝑧𝑅𝑦) ↔ (𝑧𝐴𝑧𝑅𝑤)))
148, 3, 11, 13opelopab 4957 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)} ↔ (𝑧𝐴𝑧𝑅𝑤))
157, 14bitr4i 267 . 2 (⟨𝑧, 𝑤⟩ ∈ (𝑅𝐴) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)})
161, 2, 15eqrelriiv 5175 1 (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613  {copab 4672  cres 5076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-res 5086
This theorem is referenced by:  shftidt2  13755
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