Theorem funoprab 4011

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 983	. 2 |- A.xA.yE*zph
3		funoprabg 4010	. 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 954  E*wmo 1381  Fun wfun 3176  {copab2 3964

This theorem is referenced by:  oprabex 4019  oprabex2g 4020  oprabvalig 4024  th3qcor 4316  axaddopr 5265  axmulopr 5266  oprabvaligg 10440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-fun 3192  df-oprab 3966

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