Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmoprab GIF version

Theorem dmoprab 5527
 Description: The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Assertion
Ref Expression
dmoprab dom {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨x, y⟩ ∣ zφ}
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem dmoprab
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 5494 . . 3 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
21dmeqi 4479 . 2 dom {⟨⟨x, y⟩, z⟩ ∣ φ} = dom {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
3 dmopab 4489 . 2 dom {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} = {wzxy(w = ⟨x, y φ)}
4 exrot3 1577 . . . . 5 (zxy(w = ⟨x, y φ) ↔ xyz(w = ⟨x, y φ))
5 19.42v 1783 . . . . . 6 (z(w = ⟨x, y φ) ↔ (w = ⟨x, y zφ))
652exbii 1494 . . . . 5 (xyz(w = ⟨x, y φ) ↔ xy(w = ⟨x, y zφ))
74, 6bitri 173 . . . 4 (zxy(w = ⟨x, y φ) ↔ xy(w = ⟨x, y zφ))
87abbii 2150 . . 3 {wzxy(w = ⟨x, y φ)} = {wxy(w = ⟨x, y zφ)}
9 df-opab 3810 . . 3 {⟨x, y⟩ ∣ zφ} = {wxy(w = ⟨x, y zφ)}
108, 9eqtr4i 2060 . 2 {wzxy(w = ⟨x, y φ)} = {⟨x, y⟩ ∣ zφ}
112, 3, 103eqtri 2061 1 dom {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨x, y⟩ ∣ zφ}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378  {cab 2023  ⟨cop 3370  {copab 3808  dom cdm 4288  {coprab 5456 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-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  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-dm 4298  df-oprab 5459 This theorem is referenced by:  dmoprabss  5528  reldmoprab  5531  fnoprabg  5544  dmaddpq  6363  dmmulpq  6364
 Copyright terms: Public domain W3C validator