Theorem fnopab 6631

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)

Hypotheses

Ref	Expression
fnopab.1	⊢ (𝑥 ∈ 𝐴 → ∃!𝑦𝜑)
fnopab.2	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

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

Step	Hyp	Ref	Expression
1		fnopab.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃!𝑦𝜑)
2	1	rgen 3054	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑
3		fnopab.2	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	fnopabg 6630	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴)
5	2, 4	mpbi 230	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  {copab 5161   Fn wfn 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-fun 6495  df-fn 6496

This theorem is referenced by:  fvopab3g  6937