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Theorem resopab 4595
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
Assertion
Ref Expression
resopab ({⟨x, y⟩ ∣ φ} ↾ A) = {⟨x, y⟩ ∣ (x A φ)}
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem resopab
StepHypRef Expression
1 df-res 4300 . 2 ({⟨x, y⟩ ∣ φ} ↾ A) = ({⟨x, y⟩ ∣ φ} ∩ (A × V))
2 df-xp 4294 . . . . . 6 (A × V) = {⟨x, y⟩ ∣ (x A y V)}
3 vex 2554 . . . . . . . 8 y V
43biantru 286 . . . . . . 7 (x A ↔ (x A y V))
54opabbii 3815 . . . . . 6 {⟨x, y⟩ ∣ x A} = {⟨x, y⟩ ∣ (x A y V)}
62, 5eqtr4i 2060 . . . . 5 (A × V) = {⟨x, y⟩ ∣ x A}
76ineq2i 3129 . . . 4 ({⟨x, y⟩ ∣ φ} ∩ (A × V)) = ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ x A})
8 incom 3123 . . . 4 ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ x A}) = ({⟨x, y⟩ ∣ x A} ∩ {⟨x, y⟩ ∣ φ})
97, 8eqtri 2057 . . 3 ({⟨x, y⟩ ∣ φ} ∩ (A × V)) = ({⟨x, y⟩ ∣ x A} ∩ {⟨x, y⟩ ∣ φ})
10 inopab 4411 . . 3 ({⟨x, y⟩ ∣ x A} ∩ {⟨x, y⟩ ∣ φ}) = {⟨x, y⟩ ∣ (x A φ)}
119, 10eqtri 2057 . 2 ({⟨x, y⟩ ∣ φ} ∩ (A × V)) = {⟨x, y⟩ ∣ (x A φ)}
121, 11eqtri 2057 1 ({⟨x, y⟩ ∣ φ} ↾ A) = {⟨x, y⟩ ∣ (x A φ)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  {copab 3808   × cxp 4286  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-bndl 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-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-opab 3810  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  resopab2  4598  opabresid  4602  mptpreima  4757  isarep2  4929  resoprab  5539  df1st2  5782  df2nd2  5783
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