ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnopab GIF version

Theorem fnopab 5183
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
Hypotheses
Ref Expression
fnopab.1 (𝑥𝐴 → ∃!𝑦𝜑)
fnopab.2 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Assertion
Ref Expression
fnopab 𝐹 Fn 𝐴
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fnopab
StepHypRef Expression
1 fnopab.1 . . 3 (𝑥𝐴 → ∃!𝑦𝜑)
21rgen 2444 . 2 𝑥𝐴 ∃!𝑦𝜑
3 fnopab.2 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
43fnopabg 5182 . 2 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
52, 4mpbi 144 1 𝐹 Fn 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1299  wcel 1448  ∃!weu 1960  wral 2375  {copab 3928   Fn wfn 5054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-fun 5061  df-fn 5062
This theorem is referenced by:  fvopab3g  5426
  Copyright terms: Public domain W3C validator