Theorem fnoprab 5993

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⟩ ∣ (𝜑 ∧ 𝜓)} 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

Step	Hyp	Ref	Expression
1		fnoprab.1	. . 3 ⊢ (𝜑 → ∃!𝑧𝜓)
2	1	gen2 1460	. 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
3		fnoprabg 5991	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4	2, 3	ax-mp 5	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1361  ∃!weu 2037  {copab 4077   Fn wfn 5225  {coprab 5891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-fun 5232  df-fn 5233  df-oprab 5894

This theorem is referenced by:  ovid  6007  ov  6010