Theorem fununiq 31667

Description: The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Hypotheses

Ref	Expression
fununiq.1	|- A e. _V
fununiq.2	|- B e. _V
fununiq.3	|- C e. _V

Assertion

Ref	Expression
fununiq	|- ( Fun F -> ( ( A F B /\ A F C ) -> B = C ) )

Proof of Theorem fununiq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun2 5898	. 2 |- ( Fun F <-> ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) )
2		fununiq.1	. . . 4 |- A e. _V
3		fununiq.2	. . . 4 |- B e. _V
4		fununiq.3	. . . 4 |- C e. _V
5		breq12 4658	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( x F y <-> A F B ) )
6	5	3adant3 1081	. . . . . . 7 |- ( ( x = A /\ y = B /\ z = C ) -> ( x F y <-> A F B ) )
7		breq12 4658	. . . . . . . 8 |- ( ( x = A /\ z = C ) -> ( x F z <-> A F C ) )
8	7	3adant2 1080	. . . . . . 7 |- ( ( x = A /\ y = B /\ z = C ) -> ( x F z <-> A F C ) )
9	6, 8	anbi12d 747	. . . . . 6 |- ( ( x = A /\ y = B /\ z = C ) -> ( ( x F y /\ x F z ) <-> ( A F B /\ A F C ) ) )
10		eqeq12 2635	. . . . . . 7 |- ( ( y = B /\ z = C ) -> ( y = z <-> B = C ) )
11	10	3adant1 1079	. . . . . 6 |- ( ( x = A /\ y = B /\ z = C ) -> ( y = z <-> B = C ) )
12	9, 11	imbi12d 334	. . . . 5 |- ( ( x = A /\ y = B /\ z = C ) -> ( ( ( x F y /\ x F z ) -> y = z ) <-> ( ( A F B /\ A F C ) -> B = C ) ) )
13	12	spc3gv 3298	. . . 4 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) -> ( ( A F B /\ A F C ) -> B = C ) ) )
14	2, 3, 4, 13	mp3an 1424	. . 3 |- ( A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) -> ( ( A F B /\ A F C ) -> B = C ) )
15	14	adantl 482	. 2 |- ( ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) -> ( ( A F B /\ A F C ) -> B = C ) )
16	1, 15	sylbi 207	1 |- ( Fun F -> ( ( A F B /\ A F C ) -> B = C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  funbreq  31668