Theorem eufnfv 5410

Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Hypotheses

Ref	Expression
eufnfv.1	|- A e. _V
eufnfv.2	|- B e. _V

Assertion

Ref	Expression
eufnfv	|- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )

Distinct variable groups:   x, f, A   B, f

Allowed substitution hint:   B( x)

Proof of Theorem eufnfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eufnfv.1	. . . . 5 |- A e. _V
2	1	mptex 5408	. . . 4 |- ( x e. A |-> B ) e. _V
3		eqeq2 2090	. . . . . 6 |- ( y = ( x e. A |-> B ) -> ( f = y <-> f = ( x e. A |-> B ) ) )
4	3	bibi2d 230	. . . . 5 |- ( y = ( x e. A |-> B ) -> ( ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = y ) <-> ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) ) ) )
5	4	albidv 1745	. . . 4 |- ( y = ( x e. A |-> B ) -> ( A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = y ) <-> A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) ) ) )
6	2, 5	spcev 2692	. . 3 |- ( A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) ) -> E. y A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = y ) )
7		eufnfv.2	. . . . . . 7 |- B e. _V
8		eqid 2081	. . . . . . 7 |- ( x e. A |-> B ) = ( x e. A |-> B )
9	7, 8	fnmpti 5047	. . . . . 6 |- ( x e. A |-> B ) Fn A
10		fneq1 5007	. . . . . 6 |- ( f = ( x e. A |-> B ) -> ( f Fn A <-> ( x e. A |-> B ) Fn A ) )
11	9, 10	mpbiri 166	. . . . 5 |- ( f = ( x e. A |-> B ) -> f Fn A )
12	11	pm4.71ri 384	. . . 4 |- ( f = ( x e. A |-> B ) <-> ( f Fn A /\ f = ( x e. A |-> B ) ) )
13		dffn5im 5240	. . . . . . 7 |- ( f Fn A -> f = ( x e. A |-> ( f ` x ) ) )
14	13	eqeq1d 2089	. . . . . 6 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) ) )
15		funfvex 5212	. . . . . . . . 9 |- ( ( Fun f /\ x e. dom f ) -> ( f ` x ) e. _V )
16	15	funfni 5019	. . . . . . . 8 |- ( ( f Fn A /\ x e. A ) -> ( f ` x ) e. _V )
17	16	ralrimiva 2434	. . . . . . 7 |- ( f Fn A -> A. x e. A ( f ` x ) e. _V )
18		mpteqb 5282	. . . . . . 7 |- ( A. x e. A ( f ` x ) e. _V -> ( ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
19	17, 18	syl 14	. . . . . 6 |- ( f Fn A -> ( ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
20	14, 19	bitrd 186	. . . . 5 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
21	20	pm5.32i 441	. . . 4 |- ( ( f Fn A /\ f = ( x e. A |-> B ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) = B ) )
22	12, 21	bitr2i 183	. . 3 |- ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) )
23	6, 22	mpg 1380	. 2 |- E. y A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = y )
24		df-eu 1944	. 2 |- ( E! f ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> E. y A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = y ) )
25	23, 24	mpbir 144	1 |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   A.wral 2348   _Vcvv 2601   |-> cmpt 3839   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by: (None)