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Theorem funoprabg 5542
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
funoprabg (xy∃*zφ → Fun {⟨⟨x, y⟩, z⟩ ∣ φ})
Distinct variable group:   x,y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem funoprabg
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 mosubopt 4348 . . 3 (xy∃*zφ∃*zxy(w = ⟨x, y φ))
21alrimiv 1751 . 2 (xy∃*zφw∃*zxy(w = ⟨x, y φ))
3 dfoprab2 5494 . . . 4 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
43funeqi 4865 . . 3 (Fun {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ Fun {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)})
5 funopab 4878 . . 3 (Fun {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} ↔ w∃*zxy(w = ⟨x, y φ))
64, 5bitr2i 174 . 2 (w∃*zxy(w = ⟨x, y φ) ↔ Fun {⟨⟨x, y⟩, z⟩ ∣ φ})
72, 6sylib 127 1 (xy∃*zφ → Fun {⟨⟨x, y⟩, z⟩ ∣ φ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378  ∃*wmo 1898  cop 3370  {copab 3808  Fun wfun 4839  {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-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-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847  df-oprab 5459
This theorem is referenced by:  funoprab  5543  fnoprabg  5544  oprabexd  5696
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