Theorem fv3 2839

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 group(s):   x,y,F   x,A,y

Proof of Theorem fv3

Step	Hyp	Ref	Expression
1		fv3.1	. . . 4 |- A e. V
2	1	elfv 2830	. . 3 |- (x e. (F` A) <-> E.z(x e. z /\ A.y(AFy <-> y = z)))
3		bi2 131	. . . . . . . . . 10 |- ((AFy <-> y = z) -> (y = z -> AFy))
4	3	19.20i 691	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.y(y = z -> AFy))
5		visset 1350	. . . . . . . . . 10 |- z e. V
6		breq2 2066	. . . . . . . . . 10 |- (y = z -> (AFy <-> AFz))
7	5, 6	ceqsalv 1364	. . . . . . . . 9 |- (A.y(y = z -> AFy) <-> AFz)
8	4, 7	sylib 173	. . . . . . . 8 |- (A.y(AFy <-> y = z) -> AFz)
9	8	anim2i 270	. . . . . . 7 |- ((x e. z /\ A.y(AFy <-> y = z)) -> (x e. z /\ AFz))
10	9	19.22i 723	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.z(x e. z /\ AFz))
11		eleq2 1150	. . . . . . . 8 |- (z = y -> (x e. z <-> x e. y))
12		breq2 2066	. . . . . . . 8 |- (z = y -> (AFz <-> AFy))
13	11, 12	anbi12d 476	. . . . . . 7 |- (z = y -> ((x e. z /\ AFz) <-> (x e. y /\ AFy)))
14	13	cbvexv 973	. . . . . 6 |- (E.z(x e. z /\ AFz) <-> E.y(x e. y /\ AFy))
15	10, 14	sylib 173	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.y(x e. y /\ AFy))
16		19.40 773	. . . . . . 7 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.z x e. z /\ E.zA.y(AFy <-> y = z)))
17	16	pm3.27d 262	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.zA.y(AFy <-> y = z))
18		df-eu 1009	. . . . . 6 |- (E!y AFy <-> E.zA.y(AFy <-> y = z))
19	17, 18	sylibr 175	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E!y AFy)
20	15, 19	jca 236	. . . 4 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.y(x e. y /\ AFy) /\ E!y AFy))
21		hbeu1 1015	. . . . . . 7 |- (E!y AFy -> A.yE!y AFy)
22		ax-17 925	. . . . . . . . 9 |- (x e. z -> A.y x e. z)
23		hba1 698	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.yA.y(AFy <-> y = z))
24	22, 23	hban 704	. . . . . . . 8 |- ((x e. z /\ A.y(AFy <-> y = z)) -> A.y(x e. z /\ A.y(AFy <-> y = z)))
25	24	hbex 701	. . . . . . 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 702	. . . . . 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 130	. . . . . . . . . . . . . 14 |- ((AFy <-> y = z) -> (AFy -> y = z))
28		ax-14 805	. . . . . . . . . . . . . 14 |- (y = z -> (x e. y -> x e. z))
29	27, 28	syl6 23	. . . . . . . . . . . . 13 |- ((AFy <-> y = z) -> (AFy -> (x e. y -> x e. z)))
30	29	com23 32	. . . . . . . . . . . 12 |- ((AFy <-> y = z) -> (x e. y -> (AFy -> x e. z)))
31	30	imp3a 279	. . . . . . . . . . 11 |- ((AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
32	31	a4s 682	. . . . . . . . . 10 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
33	32	anc2ri 251	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> (x e. z /\ A.y(AFy <-> y = z))))
34	33	com12 13	. . . . . . . 8 |- ((x e. y /\ AFy) -> (A.y(AFy <-> y = z) -> (x e. z /\ A.y(AFy <-> y = z))))
35	34	19.22dv 947	. . . . . . 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 181	. . . . . 6 |- ((x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
37	26, 36	19.23ai 746	. . . . 5 |- (E.y(x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
38	37	imp 277	. . . 4 |- ((E.y(x e. y /\ AFy) /\ E!y AFy) -> E.z(x e. z /\ A.y(AFy <-> y = z)))
39	20, 38	impbi 139	. . 3 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
40	2, 39	bitr 151	. 2 |- (x e. (F` A) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
41	40	biabri 1180	1 |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  E!weu 1007  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  ` cfv 2422

This theorem is referenced by:  tz6.12-1 2842  tz6.12-2 2845

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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