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Theorem rnoprab 5510
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 5475 . . 3 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
21rneqi 4489 . 2 ran {⟨⟨x, y⟩, z⟩ ∣ φ} = ran {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
3 rnopab 4508 . 2 ran {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} = {zwxy(w = ⟨x, y φ)}
4 exrot3 1562 . . . 4 (wxy(w = ⟨x, y φ) ↔ xyw(w = ⟨x, y φ))
5 vex 2538 . . . . . . . 8 x V
6 vex 2538 . . . . . . . 8 y V
75, 6opex 3940 . . . . . . 7 x, y V
87isseti 2541 . . . . . 6 w w = ⟨x, y
9 19.41v 1764 . . . . . 6 (w(w = ⟨x, y φ) ↔ (w w = ⟨x, y φ))
108, 9mpbiran 835 . . . . 5 (w(w = ⟨x, y φ) ↔ φ)
11102exbii 1479 . . . 4 (xyw(w = ⟨x, y φ) ↔ xyφ)
124, 11bitri 173 . . 3 (wxy(w = ⟨x, y φ) ↔ xyφ)
1312abbii 2135 . 2 {zwxy(w = ⟨x, y φ)} = {zxyφ}
142, 3, 133eqtri 2046 1 ran {⟨⟨x, y⟩, z⟩ ∣ φ} = {zxyφ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362  {cab 2008  cop 3353  {copab 3791  ran crn 4273  {coprab 5437
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283  df-oprab 5440
This theorem is referenced by:  rnoprab2  5511
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