Theorem eufnfv 4771

Description: A function is uniquely determined by its values.

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

Proof of Theorem eufnfv

Step	Hyp	Ref	Expression
1		eufnfv.1	. . . . 5 |- A e. _V
2	1	opabex2 4539	. . . 4 |- {<.x, y>. | (x e. A /\ y = B)} e. _V
3		eqeq2 1893	. . . . . 6 |- (z = {<.x, y>. | (x e. A /\ y = B)} -> (f = z <-> f = {<.x, y>. | (x e. A /\ y = B)}))
4	3	bibi2d 680	. . . . 5 |- (z = {<.x, y>. | (x e. A /\ y = 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, y>. | (x e. A /\ y = B)})))
5	4	albidv 1656	. . . 4 |- (z = {<.x, y>. | (x e. A /\ y = 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, y>. | (x e. A /\ y = B)})))
6	2, 5	cla4ev 2371	. . 3 |- (A.f((f Fn A /\ A.x e. A (f` x) = B) <-> f = {<.x, y>. | (x e. A /\ y = B)}) -> E.zA.f((f Fn A /\ A.x e. A (f` x) = B) <-> f = z))
7		eufnfv.2	. . . . . . 7 |- B e. _V
8		eqid 1884	. . . . . . 7 |- {<.x, y>. | (x e. A /\ y = B)} = {<.x, y>. | (x e. A /\ y = B)}
9	7, 8	fnopab2 4549	. . . . . 6 |- {<.x, y>. | (x e. A /\ y = B)} Fn A
10		fneq1 4503	. . . . . 6 |- (f = {<.x, y>. | (x e. A /\ y = B)} -> (f Fn A <-> {<.x, y>. | (x e. A /\ y = B)} Fn A))
11	9, 10	mpbiri 211	. . . . 5 |- (f = {<.x, y>. | (x e. A /\ y = B)} -> f Fn A)
12	11	pm4.71ri 700	. . . 4 |- (f = {<.x, y>. | (x e. A /\ y = B)} <-> (f Fn A /\ f = {<.x, y>. | (x e. A /\ y = B)}))
13		ax-17 1317	. . . . . . . 8 |- (z e. f -> A.x z e. f)
14		hbopab1 3562	. . . . . . . 8 |- (z e. {<.x, y>. | (x e. A /\ y = B)} -> A.x z e. {<.x, y>. | (x e. A /\ y = B)})
15	13, 14	eqfnfv2f 4770	. . . . . . 7 |- ((f Fn A /\ {<.x, y>. | (x e. A /\ y = B)} Fn A) -> (f = {<.x, y>. | (x e. A /\ y = B)} <-> A.x e. A (f` x) = ({<.x, y>. | (x e. A /\ y = B)}` x)))
16	9, 15	mpan2 760	. . . . . 6 |- (f Fn A -> (f = {<.x, y>. | (x e. A /\ y = B)} <-> A.x e. A (f` x) = ({<.x, y>. | (x e. A /\ y = B)}` x)))
17		fvopab2 4754	. . . . . . . . 9 |- ((x e. A /\ B e. _V) -> ({<.x, y>. | (x e. A /\ y = B)}` x) = B)
18	7, 17	mpan2 760	. . . . . . . 8 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = B)}` x) = B)
19	18	eqeq2d 1895	. . . . . . 7 |- (x e. A -> ((f` x) = ({<.x, y>. | (x e. A /\ y = B)}` x) <-> (f` x) = B))
20	19	ralbiia 2133	. . . . . 6 |- (A.x e. A (f` x) = ({<.x, y>. | (x e. A /\ y = B)}` x) <-> A.x e. A (f` x) = B)
21	16, 20	syl6bb 595	. . . . 5 |- (f Fn A -> (f = {<.x, y>. | (x e. A /\ y = B)} <-> A.x e. A (f` x) = B))
22	21	pm5.32i 707	. . . 4 |- ((f Fn A /\ f = {<.x, y>. | (x e. A /\ y = B)}) <-> (f Fn A /\ A.x e. A (f` x) = B))
23	12, 22	bitr2i 191	. . 3 |- ((f Fn A /\ A.x e. A (f` x) = B) <-> f = {<.x, y>. | (x e. A /\ y = B)})
24	6, 23	mpg 1332	. 2 |- E.zA.f((f Fn A /\ A.x e. A (f` x) = B) <-> f = z)
25		df-eu 1775	. 2 |- (E!f(f Fn A /\ A.x e. A (f` x) = B) <-> E.zA.f((f Fn A /\ A.x e. A (f` x) = B) <-> f = z))
26	24, 25	mpbir 207	1 |- E!f(f Fn A /\ A.x e. A (f` x) = B)

Syntax hints:   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  A.wral 2105  _Vcvv 2292  {copab 3395   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  eustrdif 16722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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