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Theorem fnopab 5023
 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 2374 . 2 𝑥𝐴 ∃!𝑦𝜑
3 fnopab.2 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
43fnopabg 5022 . 2 (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
52, 4mpbi 133 1 𝐹 Fn 𝐴
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∃!weu 1900  ∀wral 2306  {copab 3817   Fn wfn 4897 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-fn 4905 This theorem is referenced by:  fvopab3g  5245
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