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Theorem fnopab 6640
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 3063 . 2 𝑥𝐴 ∃!𝑦𝜑
3 fnopab.2 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
43fnopabg 6639 . 2 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
52, 4mpbi 229 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ∃!weu 2563  wral 3061  {copab 5168   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500
This theorem is referenced by:  fvopab3g  6944
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