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Theorem dfres2 4601
Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
Assertion
Ref Expression
dfres2 (𝑅A) = {⟨x, y⟩ ∣ (x A x𝑅y)}
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 4582 . 2 Rel (𝑅A)
2 relopab 4407 . 2 Rel {⟨x, y⟩ ∣ (x A x𝑅y)}
3 vex 2554 . . . . 5 w V
43brres 4561 . . . 4 (z(𝑅A)w ↔ (z𝑅w z A))
5 df-br 3756 . . . 4 (z(𝑅A)w ↔ ⟨z, w (𝑅A))
6 ancom 253 . . . 4 ((z𝑅w z A) ↔ (z A z𝑅w))
74, 5, 63bitr3i 199 . . 3 (⟨z, w (𝑅A) ↔ (z A z𝑅w))
8 vex 2554 . . . 4 z V
9 eleq1 2097 . . . . 5 (x = z → (x Az A))
10 breq1 3758 . . . . 5 (x = z → (x𝑅yz𝑅y))
119, 10anbi12d 442 . . . 4 (x = z → ((x A x𝑅y) ↔ (z A z𝑅y)))
12 breq2 3759 . . . . 5 (y = w → (z𝑅yz𝑅w))
1312anbi2d 437 . . . 4 (y = w → ((z A z𝑅y) ↔ (z A z𝑅w)))
148, 3, 11, 13opelopab 3999 . . 3 (⟨z, w {⟨x, y⟩ ∣ (x A x𝑅y)} ↔ (z A z𝑅w))
157, 14bitr4i 176 . 2 (⟨z, w (𝑅A) ↔ ⟨z, w {⟨x, y⟩ ∣ (x A x𝑅y)})
161, 2, 15eqrelriiv 4377 1 (𝑅A) = {⟨x, y⟩ ∣ (x A x𝑅y)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  {copab 3808  cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by: (None)
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