Theorem eufnfv 6156

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 z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eufnfv.1	. . . . 5 |- A e. _V
2	1	mptex 6152	. . . 4 |- ( x e. A |-> B ) e. _V
3		eqeq2 2482	. . . . . 6 |- ( z = ( x e. A |-> B ) -> ( f = z <-> f = ( x e. A |-> B ) ) )
4	3	bibi2d 325	. . . . 5 |- ( z = ( x e. A |-> B ) -> ( ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z ) <-> ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) ) ) )
5	4	albidv 1775	. . . 4 |- ( z = ( x e. A |-> B ) -> ( A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z ) <-> A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) ) ) )
6	2, 5	spcev 3127	. . 3 |- ( A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) ) -> E. z A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z ) )
7		eufnfv.2	. . . . . . 7 |- B e. _V
8		eqid 2471	. . . . . . 7 |- ( x e. A |-> B ) = ( x e. A |-> B )
9	7, 8	fnmpti 5716	. . . . . 6 |- ( x e. A |-> B ) Fn A
10		fneq1 5674	. . . . . 6 |- ( f = ( x e. A |-> B ) -> ( f Fn A <-> ( x e. A |-> B ) Fn A ) )
11	9, 10	mpbiri 241	. . . . 5 |- ( f = ( x e. A |-> B ) -> f Fn A )
12	11	pm4.71ri 645	. . . 4 |- ( f = ( x e. A |-> B ) <-> ( f Fn A /\ f = ( x e. A |-> B ) ) )
13		dffn5 5924	. . . . . . 7 |- ( f Fn A <-> f = ( x e. A |-> ( f ` x ) ) )
14		eqeq1 2475	. . . . . . 7 |- ( f = ( x e. A |-> ( f ` x ) ) -> ( f = ( x e. A |-> B ) <-> ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) ) )
15	13, 14	sylbi 200	. . . . . 6 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) ) )
16		fvex 5889	. . . . . . . 8 |- ( f ` x ) e. _V
17	16	rgenw 2768	. . . . . . 7 |- A. x e. A ( f ` x ) e. _V
18		mpteqb 5979	. . . . . . 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	ax-mp 5	. . . . . 6 |- ( ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B )
20	15, 19	syl6bb 269	. . . . 5 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
21	20	pm5.32i 649	. . . 4 |- ( ( f Fn A /\ f = ( x e. A |-> B ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) = B ) )
22	12, 21	bitr2i 258	. . 3 |- ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) )
23	6, 22	mpg 1679	. 2 |- E. z A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z )
24		df-eu 2323	. 2 |- ( E! f ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> E. z A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z ) )
25	23, 24	mpbir 214	1 |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )

Syntax hints:   <-> wb 189   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   E!weu 2319   A.wral 2756   _Vcvv 3031   |-> cmpt 4454   Fn wfn 5584   ` cfv 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597

This theorem is referenced by: (None)