Theorem fnoprab 7539

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 1829	. 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓)
3		fnoprabg 7537	. 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}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃!weu 2593  {copab 5167   Fn wfn 6528  {coprab 7415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-fun 6535  df-fn 6536  df-oprab 7418

This theorem is used by:  ovid  7555  ov  7558