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Theorem funopab 4850
Description: A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
Assertion
Ref Expression
funopab (Fun {⟨x, y⟩ ∣ φ} ↔ x∃*yφ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem funopab
StepHypRef Expression
1 relopab 4380 . . 3 Rel {⟨x, y⟩ ∣ φ}
2 nfopab1 3790 . . . 4 x{⟨x, y⟩ ∣ φ}
3 nfopab2 3791 . . . 4 y{⟨x, y⟩ ∣ φ}
42, 3dffun6f 4830 . . 3 (Fun {⟨x, y⟩ ∣ φ} ↔ (Rel {⟨x, y⟩ ∣ φ} x∃*y x{⟨x, y⟩ ∣ φ}y))
51, 4mpbiran 829 . 2 (Fun {⟨x, y⟩ ∣ φ} ↔ x∃*y x{⟨x, y⟩ ∣ φ}y)
6 df-br 3729 . . . . 5 (x{⟨x, y⟩ ∣ φ}y ↔ ⟨x, y {⟨x, y⟩ ∣ φ})
7 opabid 3958 . . . . 5 (⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)
86, 7bitri 173 . . . 4 (x{⟨x, y⟩ ∣ φ}yφ)
98mobii 1911 . . 3 (∃*y x{⟨x, y⟩ ∣ φ}y∃*yφ)
109albii 1333 . 2 (x∃*y x{⟨x, y⟩ ∣ φ}yx∃*yφ)
115, 10bitri 173 1 (Fun {⟨x, y⟩ ∣ φ} ↔ x∃*yφ)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1222   wcel 1367  ∃*wmo 1875  cop 3343   class class class wbr 3728  {copab 3781  Rel wrel 4266  Fun wfun 4812
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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-fun 4820
This theorem is referenced by:  funopabeq  4851  isarep2  4901  fnopabg  4937  fvopab3ig  5160  opabex  5299  funoprabg  5512
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