Theorem eufnfv 5870

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 5865	. . . 4 |- ( x e. A |-> B ) e. _V
3		eqeq2 2239	. . . . . 6 |- ( y = ( x e. A |-> B ) -> ( f = y <-> f = ( x e. A |-> B ) ) )
4	3	bibi2d 232	. . . . 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 1870	. . . 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 2898	. . 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 2229	. . . . . . 7 |- ( x e. A |-> B ) = ( x e. A |-> B )
9	7, 8	fnmpti 5452	. . . . . 6 |- ( x e. A |-> B ) Fn A
10		fneq1 5409	. . . . . 6 |- ( f = ( x e. A |-> B ) -> ( f Fn A <-> ( x e. A |-> B ) Fn A ) )
11	9, 10	mpbiri 168	. . . . 5 |- ( f = ( x e. A |-> B ) -> f Fn A )
12	11	pm4.71ri 392	. . . 4 |- ( f = ( x e. A |-> B ) <-> ( f Fn A /\ f = ( x e. A |-> B ) ) )
13		dffn5im 5679	. . . . . . 7 |- ( f Fn A -> f = ( x e. A |-> ( f ` x ) ) )
14	13	eqeq1d 2238	. . . . . 6 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) ) )
15		funfvex 5644	. . . . . . . . 9 |- ( ( Fun f /\ x e. dom f ) -> ( f ` x ) e. _V )
16	15	funfni 5423	. . . . . . . 8 |- ( ( f Fn A /\ x e. A ) -> ( f ` x ) e. _V )
17	16	ralrimiva 2603	. . . . . . 7 |- ( f Fn A -> A. x e. A ( f ` x ) e. _V )
18		mpteqb 5725	. . . . . . 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 188	. . . . 5 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
21	20	pm5.32i 454	. . . 4 |- ( ( f Fn A /\ f = ( x e. A |-> B ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) = B ) )
22	12, 21	bitr2i 185	. . 3 |- ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) )
23	6, 22	mpg 1497	. 2 |- E. y A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = y )
24		df-eu 2080	. 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 146	1 |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   A.wral 2508   _Vcvv 2799   |-> cmpt 4145   Fn wfn 5313   ` cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326

This theorem is referenced by: (None)