Theorem funopab 2694

Description: A class of ordered pairs is a function when there is at most one second member for each pair.

Assertion

Ref	Expression
funopab	⊢ (Fun {⟨x, y⟩∣φ} ↔ ∀x∃*yφ)

Distinct variable group(s):   x,y

Proof of Theorem funopab

Step	Hyp	Ref	Expression
1		hbopab1 2112	. . . 4 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀x z ∈ {⟨x, y⟩∣φ})
2		hbopab2 2113	. . . 4 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})
3	1, 2	dffunmof 2678	. . 3 ⊢ (Fun {⟨x, y⟩∣φ} ↔ (Rel {⟨x, y⟩∣φ} ∧ ∀x∃*y x{⟨x, y⟩∣φ}y))
4		relopab 2494	. . 3 ⊢ Rel {⟨x, y⟩∣φ}
5	3, 4	mpbiran 547	. 2 ⊢ (Fun {⟨x, y⟩∣φ} ↔ ∀x∃*y x{⟨x, y⟩∣φ}y)
6		df-br 2063	. . . . 5 ⊢ (x{⟨x, y⟩∣φ}y ↔ ⟨x, y⟩ ∈ {⟨x, y⟩∣φ})
7		opabid 2099	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)
8	6, 7	bitr 151	. . . 4 ⊢ (x{⟨x, y⟩∣φ}y ↔ φ)
9	8	bimo 1031	. . 3 ⊢ (∃*y x{⟨x, y⟩∣φ}y ↔ ∃*yφ)
10	9	bial 695	. 2 ⊢ (∀x∃*y x{⟨x, y⟩∣φ}y ↔ ∀x∃*yφ)
11	5, 10	bitr 151	1 ⊢ (Fun {⟨x, y⟩∣φ} ↔ ∀x∃*yφ)

Syntax hints:   ↔ wb 127  ∀wal 672  ∃*wmo 1008   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  {copab 2055  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  funopabeq 2695  funopabex 2742  zfrep6 2744  fnopabg 2745  fvopab3ig 2869  tz7.44lem1 2965  funoprab 3037  qsex 3231

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

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