Theorem funoprabg 4016

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

Assertion

Ref	Expression
funoprabg	|- (A.xA.yE*zph -> Fun {<.<.x, y>., z>. | ph})

Distinct variable group:   x,y,z

Proof of Theorem funoprabg

Step	Hyp	Ref	Expression
1		mosubopt 2810	. . 3 |- (A.xA.yE*zph -> E*zE.xE.y(w = <.x, y>. /\ ph))
2	1	19.21aiv 1288	. 2 |- (A.xA.yE*zph -> A.wE*zE.xE.y(w = <.x, y>. /\ ph))
3		dfoprab2 3997	. . . 4 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
4		funeq 3541	. . . 4 |- ({<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)} -> (Fun {<.<.x, y>., z>. | ph} <-> Fun {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}))
5	3, 4	ax-mp 7	. . 3 |- (Fun {<.<.x, y>., z>. | ph} <-> Fun {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)})
6		funopab 3554	. . 3 |- (Fun {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)} <-> A.wE*zE.xE.y(w = <.x, y>. /\ ph))
7	5, 6	bitr2 174	. 2 |- (A.wE*zE.xE.y(w = <.x, y>. /\ ph) <-> Fun {<.<.x, y>., z>. | ph})
8	2, 7	sylib 198	1 |- (A.xA.yE*zph -> Fun {<.<.x, y>., z>. | ph})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958  E.wex 982  E*wmo 1383  <.cop 2415  {copab 2671  Fun wfun 3182  {copab2 3970

This theorem is referenced by:  funoprab 4017  fnoprabg 4018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-fun 3198  df-oprab 3972

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