Theorem fnoprab 7471

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 1797	. 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
3		fnoprabg 7469	. 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 395  ∀wal 1539  ∃!weu 2563  {copab 5151   Fn wfn 6476  {coprab 7347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-fun 6483  df-fn 6484  df-oprab 7350

This theorem is referenced by:  ovid  7487  ov  7490