Theorem eufnfv 6491

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 6486	. . . 4 |- ( x e. A |-> B ) e. _V
3		eqeq2 2633	. . . . . 6 |- ( z = ( x e. A |-> B ) -> ( f = z <-> f = ( x e. A |-> B ) ) )
4	3	bibi2d 332	. . . . 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 1849	. . . 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 3300	. . 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 2622	. . . . . . 7 |- ( x e. A |-> B ) = ( x e. A |-> B )
9	7, 8	fnmpti 6022	. . . . . 6 |- ( x e. A |-> B ) Fn A
10		fneq1 5979	. . . . . 6 |- ( f = ( x e. A |-> B ) -> ( f Fn A <-> ( x e. A |-> B ) Fn A ) )
11	9, 10	mpbiri 248	. . . . 5 |- ( f = ( x e. A |-> B ) -> f Fn A )
12	11	pm4.71ri 665	. . . 4 |- ( f = ( x e. A |-> B ) <-> ( f Fn A /\ f = ( x e. A |-> B ) ) )
13		dffn5 6241	. . . . . . 7 |- ( f Fn A <-> f = ( x e. A |-> ( f ` x ) ) )
14		eqeq1 2626	. . . . . . 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 207	. . . . . 6 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) ) )
16		fvex 6201	. . . . . . . 8 |- ( f ` x ) e. _V
17	16	rgenw 2924	. . . . . . 7 |- A. x e. A ( f ` x ) e. _V
18		mpteqb 6299	. . . . . . 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 276	. . . . 5 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
21	20	pm5.32i 669	. . . 4 |- ( ( f Fn A /\ f = ( x e. A |-> B ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) = B ) )
22	12, 21	bitr2i 265	. . 3 |- ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) )
23	6, 22	mpg 1724	. 2 |- E. z A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z )
24		df-eu 2474	. 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 221	1 |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   A.wral 2912   _Vcvv 3200   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)