Theorem funoprab 3037

Description: "At most one" is sufficient condition for an operation abstraction to be a function.

Hypothesis

Ref	Expression
funoprab.1	⊢ ∃*zφ

Assertion

Ref	Expression
funoprab	⊢ Fun {⟨⟨x, y⟩, z⟩∣φ}

Distinct variable group(s):   x,y,z

Proof of Theorem funoprab

Step	Hyp	Ref	Expression
1		funopab 2694	. . 3 ⊢ (Fun {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} ↔ ∀w∃*z∃x∃y(w = ⟨x, y⟩ ∧ φ))
2		funoprab.1	. . . 4 ⊢ ∃*zφ
3	2	mosubop 1911	. . 3 ⊢ ∃*z∃x∃y(w = ⟨x, y⟩ ∧ φ)
4	1, 3	mpgbir 686	. 2 ⊢ Fun {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
5		dfoprab2 3021	. . 3 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
6		funeq 2683	. . 3 ⊢ ({⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} → (Fun {⟨⟨x, y⟩, z⟩∣φ} ↔ Fun {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}))
7	5, 6	ax-mp 6	. 2 ⊢ (Fun {⟨⟨x, y⟩, z⟩∣φ} ↔ Fun {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)})
8	4, 7	mpbir 165	1 ⊢ Fun {⟨⟨x, y⟩, z⟩∣φ}

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃*wmo 1008   = wceq 1091  ⟨cop 1810  {copab 2055  Fun wfun 2416  {copab2 3002

This theorem is referenced by:  fnoprab 3038  oprabex 3044  oprabvalig 3048  th3qcor 3252

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-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-fun 2432  df-oprab 3004

metamath.org