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Theorem rnoprab 5529
Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
Assertion
Ref Expression
rnoprab ran {⟨⟨x, y⟩, z⟩ ∣ φ} = {zxyφ}
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem rnoprab
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)}
21rneqi 4505 . 2 ran {⟨⟨x, y⟩, z⟩ ∣ φ} = ran {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
3 rnopab 4524 . 2 ran {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} = {zwxy(w = ⟨x, y φ)}
4 exrot3 1577 . . . 4 (wxy(w = ⟨x, y φ) ↔ xyw(w = ⟨x, y φ))
5 vex 2554 . . . . . . . 8 x V
6 vex 2554 . . . . . . . 8 y V
75, 6opex 3957 . . . . . . 7 x, y V
87isseti 2557 . . . . . 6 w w = ⟨x, y
9 19.41v 1779 . . . . . 6 (w(w = ⟨x, y φ) ↔ (w w = ⟨x, y φ))
108, 9mpbiran 846 . . . . 5 (w(w = ⟨x, y φ) ↔ φ)
11102exbii 1494 . . . 4 (xyw(w = ⟨x, y φ) ↔ xyφ)
124, 11bitri 173 . . 3 (wxy(w = ⟨x, y φ) ↔ xyφ)
1312abbii 2150 . 2 {zwxy(w = ⟨x, y φ)} = {zxyφ}
142, 3, 133eqtri 2061 1 ran {⟨⟨x, y⟩, z⟩ ∣ φ} = {zxyφ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378  {cab 2023  cop 3370  {copab 3808  ran crn 4289  {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-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-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-cnv 4296  df-dm 4298  df-rn 4299  df-oprab 5459
This theorem is referenced by:  rnoprab2  5530
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