Theorem fv3 3740

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 3729	. . 3 |- (x e. (F` A) <-> E.z(x e. z /\ A.y(AFy <-> y = z)))
3		bi2 149	. . . . . . . . . 10 |- ((AFy <-> y = z) -> (y = z -> AFy))
4	3	19.20i 994	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.y(y = z -> AFy))
5		visset 1816	. . . . . . . . . 10 |- z e. V
6		breq2 2629	. . . . . . . . . 10 |- (y = z -> (AFy <-> AFz))
7	5, 6	ceqsalv 1830	. . . . . . . . 9 |- (A.y(y = z -> AFy) <-> AFz)
8	4, 7	sylib 198	. . . . . . . 8 |- (A.y(AFy <-> y = z) -> AFz)
9	8	anim2i 335	. . . . . . 7 |- ((x e. z /\ A.y(AFy <-> y = z)) -> (x e. z /\ AFz))
10	9	19.22i 1042	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.z(x e. z /\ AFz))
11		eleq2 1538	. . . . . . . 8 |- (z = y -> (x e. z <-> x e. y))
12		breq2 2629	. . . . . . . 8 |- (z = y -> (AFz <-> AFy))
13	11, 12	anbi12d 630	. . . . . . 7 |- (z = y -> ((x e. z /\ AFz) <-> (x e. y /\ AFy)))
14	13	cbvexv 1317	. . . . . 6 |- (E.z(x e. z /\ AFz) <-> E.y(x e. y /\ AFy))
15	10, 14	sylib 198	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.y(x e. y /\ AFy))
16		19.40 1096	. . . . . . 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 325	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.zA.y(AFy <-> y = z))
18		df-eu 1384	. . . . . 6 |- (E!y AFy <-> E.zA.y(AFy <-> y = z))
19	17, 18	sylibr 200	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E!y AFy)
20	15, 19	jca 288	. . . 4 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.y(x e. y /\ AFy) /\ E!y AFy))
21		hbeu1 1390	. . . . . . 7 |- (E!y AFy -> A.yE!y AFy)
22		ax-17 973	. . . . . . . . 9 |- (x e. z -> A.y x e. z)
23		hba1 1005	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.yA.y(AFy <-> y = z))
24	22, 23	hban 1011	. . . . . . . 8 |- ((x e. z /\ A.y(AFy <-> y = z)) -> A.y(x e. z /\ A.y(AFy <-> y = z)))
25	24	hbex 1008	. . . . . . 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 1009	. . . . . 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 148	. . . . . . . . . . . . . 14 |- ((AFy <-> y = z) -> (AFy -> y = z))
28		ax-14 972	. . . . . . . . . . . . . 14 |- (y = z -> (x e. y -> x e. z))
29	27, 28	syl6 22	. . . . . . . . . . . . 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 361	. . . . . . . . . . 11 |- ((AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
32	31	a4s 986	. . . . . . . . . 10 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
33	32	anc2ri 303	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> (x e. z /\ A.y(AFy <-> y = z))))
34	33	com12 11	. . . . . . . 8 |- ((x e. y /\ AFy) -> (A.y(AFy <-> y = z) -> (x e. z /\ A.y(AFy <-> y = z))))
35	34	19.22dv 1292	. . . . . . 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 206	. . . . . 6 |- ((x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
37	26, 36	19.23ai 1066	. . . . 5 |- (E.y(x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
38	37	imp 350	. . . 4 |- ((E.y(x e. y /\ AFy) /\ E!y AFy) -> E.z(x e. z /\ A.y(AFy <-> y = z)))
39	20, 38	impbi 157	. . 3 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
40	2, 39	bitr 173	. 2 |- (x e. (F` A) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
41	40	abbi2i 1577	1 |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  E!weu 1382  {cab 1466  Vcvv 1814   class class class wbr 2625  ` cfv 3189

This theorem is referenced by:  tz6.12-1 3743  tz6.12-2 3746

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-xp 3191  df-cnv 3193  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fv 3205

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