MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnoprab Structured version   Visualization version   GIF version

Theorem fnoprab 7392
Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
fnoprab.1 (𝜑 → ∃!𝑧𝜓)
Assertion
Ref Expression
fnoprab {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem fnoprab
StepHypRef Expression
1 fnoprab.1 . . 3 (𝜑 → ∃!𝑧𝜓)
21gen2 1803 . 2 𝑥𝑦(𝜑 → ∃!𝑧𝜓)
3 fnoprabg 7389 . 2 (∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
42, 3ax-mp 5 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1540  ∃!weu 2570  {copab 5141   Fn wfn 6426  {coprab 7270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-fun 6433  df-fn 6434  df-oprab 7273
This theorem is referenced by:  ovid  7406  ov  7409
  Copyright terms: Public domain W3C validator