Theorem fnoprab 7562

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 1794	. 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
3		fnoprabg 7560	. 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 1536  ∃!weu 2567  {copab 5211   Fn wfn 6561  {coprab 7436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-fun 6568  df-fn 6569  df-oprab 7439

This theorem is referenced by:  ovid  7578  ov  7581