Theorem fnopab 2746

Description: Functionality and domain of an ordered pair abstraction.

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(s):   x,y,A

Proof of Theorem fnopab

Step	Hyp	Ref	Expression
1		fnopab.1	. . 3 ⊢ (x ∈ A → ∃!yφ)
2	1	rgen 1247	. 2 ⊢ ∀x ∈ A ∃!yφ
3		fnopab.2	. . 3 ⊢ F = {⟨x, y⟩∣(x ∈ A ∧ φ)}
4	3	fnopabg 2745	. 2 ⊢ (∀x ∈ A ∃!yφ ↔ F Fn A)
5	2, 4	mpbi 164	1 ⊢ F Fn A

Syntax hints:   → wi 2   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {copab 2055   Fn wfn 2417

This theorem is referenced by:  fnopab2 2747  fvopab3 2868  elrnopab 2884

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433

metamath.org