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

Theorem fnopab 5322
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 2523 . 2 𝑥𝐴 ∃!𝑦𝜑
3 fnopab.2 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
43fnopabg 5321 . 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 1348  ∃!weu 2019  wcel 2141  wral 2448  {copab 4049   Fn wfn 5193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-fn 5201
This theorem is referenced by:  fvopab3g  5569
  Copyright terms: Public domain W3C validator