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Theorem funoprabg 5523
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 4332 . . 3 (xy∃*zφ∃*zxy(w = ⟨x, y φ))
21alrimiv 1736 . 2 (xy∃*zφw∃*zxy(w = ⟨x, y φ))
3 dfoprab2 5475 . . . 4 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
43funeqi 4848 . . 3 (Fun {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ Fun {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)})
5 funopab 4861 . . 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 1226   = wceq 1228  wex 1362  ∃*wmo 1883  cop 3353  {copab 3791  Fun wfun 4823  {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-ral 2289  df-rex 2290  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-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-fun 4831  df-oprab 5440
This theorem is referenced by:  funoprab  5524  fnoprabg  5525  oprabexd  5677
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