Theorem eufnfv 6121

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 6118	. . . 4 |- ( x e. A |-> B ) e. _V
3		eqeq2 2469	. . . . . 6 |- ( z = ( x e. A |-> B ) -> ( f = z <-> f = ( x e. A |-> B ) ) )
4	3	bibi2d 316	. . . . 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 1718	. . . 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 3198	. . 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 2454	. . . . . . 7 |- ( x e. A |-> B ) = ( x e. A |-> B )
9	7, 8	fnmpti 5691	. . . . . 6 |- ( x e. A |-> B ) Fn A
10		fneq1 5651	. . . . . 6 |- ( f = ( x e. A |-> B ) -> ( f Fn A <-> ( x e. A |-> B ) Fn A ) )
11	9, 10	mpbiri 233	. . . . 5 |- ( f = ( x e. A |-> B ) -> f Fn A )
12	11	pm4.71ri 631	. . . 4 |- ( f = ( x e. A |-> B ) <-> ( f Fn A /\ f = ( x e. A |-> B ) ) )
13		dffn5 5893	. . . . . . 7 |- ( f Fn A <-> f = ( x e. A |-> ( f ` x ) ) )
14		eqeq1 2458	. . . . . . 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 195	. . . . . 6 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> ( x e. A |-> ( f ` x ) ) = ( x e. A |-> B ) ) )
16		fvex 5858	. . . . . . . 8 |- ( f ` x ) e. _V
17	16	rgenw 2815	. . . . . . 7 |- A. x e. A ( f ` x ) e. _V
18		mpteqb 5946	. . . . . . 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 261	. . . . 5 |- ( f Fn A -> ( f = ( x e. A |-> B ) <-> A. x e. A ( f ` x ) = B ) )
21	20	pm5.32i 635	. . . 4 |- ( ( f Fn A /\ f = ( x e. A |-> B ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) = B ) )
22	12, 21	bitr2i 250	. . 3 |- ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = ( x e. A |-> B ) )
23	6, 22	mpg 1625	. 2 |- E. z A. f ( ( f Fn A /\ A. x e. A ( f ` x ) = B ) <-> f = z )
24		df-eu 2288	. 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 209	1 |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )

Syntax hints:   <-> wb 184   /\ wa 367   A.wal 1396   = wceq 1398   E.wex 1617   e. wcel 1823   E!weu 2284   A.wral 2804   _Vcvv 3106   |-> cmpt 4497   Fn wfn 5565   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578

This theorem is referenced by: (None)