Theorem funoprab 4017

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

Hypothesis

Ref	Expression
funoprab.1	|- E*zph

Assertion

Ref	Expression
funoprab	|- Fun {<.<.x, y>., z>. | ph}

Distinct variable group:   x,y,z

Proof of Theorem funoprab

Step	Hyp	Ref	Expression
1		funoprab.1	. . 3 |- E*zph
2	1	gen2 985	. 2 |- A.xA.yE*zph
3		funoprabg 4016	. 2 |- (A.xA.yE*zph -> Fun {<.<.x, y>., z>. | ph})
4	2, 3	ax-mp 7	1 |- Fun {<.<.x, y>., z>. | ph}

Syntax hints:  A.wal 956  E*wmo 1383  Fun wfun 3182  {copab2 3970

This theorem is referenced by:  oprabex 4025  oprabex2g 4026  oprabvalig 4030  th3qcor 4322  axaddopr 5277  axmulopr 5278  oprabvaligg 10435

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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