Theorem fv3 4690

Description: Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26.

Hypothesis

Ref	Expression
fv3.1	|- A e. _V

Assertion

Ref	Expression
fv3	|- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}

Distinct variable groups:   x,y,F   x,A,y

Proof of Theorem fv3

Step	Hyp	Ref	Expression
1		fv3.1	. . . 4 |- A e. _V
2	1	elfv 4679	. . 3 |- (x e. (F` A) <-> E.z(x e. z /\ A.y(AFy <-> y = z)))
3		bi2 166	. . . . . . . . . 10 |- ((AFy <-> y = z) -> (y = z -> AFy))
4	3	alimi 1338	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.y(y = z -> AFy))
5		visset 2295	. . . . . . . . . 10 |- z e. _V
6		breq2 3342	. . . . . . . . . 10 |- (y = z -> (AFy <-> AFz))
7	5, 6	ceqsalv 2317	. . . . . . . . 9 |- (A.y(y = z -> AFy) <-> AFz)
8	4, 7	sylib 215	. . . . . . . 8 |- (A.y(AFy <-> y = z) -> AFz)
9	8	anim2i 362	. . . . . . 7 |- ((x e. z /\ A.y(AFy <-> y = z)) -> (x e. z /\ AFz))
10	9	eximi 1387	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.z(x e. z /\ AFz))
11		eleq2 1958	. . . . . . . 8 |- (z = y -> (x e. z <-> x e. y))
12		breq2 3342	. . . . . . . 8 |- (z = y -> (AFz <-> AFy))
13	11, 12	anbi12d 690	. . . . . . 7 |- (z = y -> ((x e. z /\ AFz) <-> (x e. y /\ AFy)))
14	13	cbvexv 1697	. . . . . 6 |- (E.z(x e. z /\ AFz) <-> E.y(x e. y /\ AFy))
15	10, 14	sylib 215	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.y(x e. y /\ AFy))
16		19.40 1447	. . . . . . 7 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.z x e. z /\ E.zA.y(AFy <-> y = z)))
17	16	simprd 352	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.zA.y(AFy <-> y = z))
18		df-eu 1775	. . . . . 6 |- (E!y AFy <-> E.zA.y(AFy <-> y = z))
19	17, 18	sylibr 217	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E!y AFy)
20	15, 19	jca 310	. . . 4 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.y(x e. y /\ AFy) /\ E!y AFy))
21		hbeu1 1781	. . . . . . 7 |- (E!y AFy -> A.yE!y AFy)
22		ax-17 1317	. . . . . . . . 9 |- (x e. z -> A.y x e. z)
23		hba1 1350	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.yA.y(AFy <-> y = z))
24	22, 23	hban 1356	. . . . . . . 8 |- ((x e. z /\ A.y(AFy <-> y = z)) -> A.y(x e. z /\ A.y(AFy <-> y = z)))
25	24	hbex 1353	. . . . . . 7 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> A.yE.z(x e. z /\ A.y(AFy <-> y = z)))
26	21, 25	hbim 1354	. . . . . 6 |- ((E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))) -> A.y(E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
27		bi1 165	. . . . . . . . . . . . . 14 |- ((AFy <-> y = z) -> (AFy -> y = z))
28		ax-14 1312	. . . . . . . . . . . . . 14 |- (y = z -> (x e. y -> x e. z))
29	27, 28	syl6 25	. . . . . . . . . . . . 13 |- ((AFy <-> y = z) -> (AFy -> (x e. y -> x e. z)))
30	29	com23 36	. . . . . . . . . . . 12 |- ((AFy <-> y = z) -> (x e. y -> (AFy -> x e. z)))
31	30	imp3a 388	. . . . . . . . . . 11 |- ((AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
32	31	a4s 1330	. . . . . . . . . 10 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
33	32	anc2ri 327	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> (x e. z /\ A.y(AFy <-> y = z))))
34	33	com12 14	. . . . . . . 8 |- ((x e. y /\ AFy) -> (A.y(AFy <-> y = z) -> (x e. z /\ A.y(AFy <-> y = z))))
35	34	eximdv 1669	. . . . . . 7 |- ((x e. y /\ AFy) -> (E.zA.y(AFy <-> y = z) -> E.z(x e. z /\ A.y(AFy <-> y = z))))
36	35, 18	syl5ib 223	. . . . . 6 |- ((x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
37	26, 36	19.23ai 1412	. . . . 5 |- (E.y(x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
38	37	imp 377	. . . 4 |- ((E.y(x e. y /\ AFy) /\ E!y AFy) -> E.z(x e. z /\ A.y(AFy <-> y = z)))
39	20, 38	impbii 174	. . 3 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
40	2, 39	bitri 190	. 2 |- (x e. (F` A) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
41	40	abbi2i 2005	1 |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  _Vcvv 2292   class class class wbr 3338  ` cfv 3998

This theorem is referenced by:  tz6.12-1 4693  tz6.12-2 4696

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

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-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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014

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