Theorem fv3 3672

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 3661	. . 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 968	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.y(y = z -> AFy))
5		visset 1788	. . . . . . . . . 10 |- z e. V
6		breq2 2591	. . . . . . . . . 10 |- (y = z -> (AFy <-> AFz))
7	5, 6	ceqsalv 1802	. . . . . . . . 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 1016	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.z(x e. z /\ AFz))
11		eleq2 1511	. . . . . . . 8 |- (z = y -> (x e. z <-> x e. y))
12		breq2 2591	. . . . . . . 8 |- (z = y -> (AFz <-> AFy))
13	11, 12	anbi12d 626	. . . . . . 7 |- (z = y -> ((x e. z /\ AFz) <-> (x e. y /\ AFy)))
14	13	cbvexv 1297	. . . . . 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 1070	. . . . . . 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 1359	. . . . . 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 1365	. . . . . . 7 |- (E!y AFy -> A.yE!y AFy)
22		ax-17 1190	. . . . . . . . 9 |- (x e. z -> A.y x e. z)
23		hba1 979	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.yA.y(AFy <-> y = z))
24	22, 23	hban 985	. . . . . . . 8 |- ((x e. z /\ A.y(AFy <-> y = z)) -> A.y(x e. z /\ A.y(AFy <-> y = z)))
25	24	hbex 982	. . . . . . 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 983	. . . . . 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 1108	. . . . . . . . . . . . . 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 960	. . . . . . . . . 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 1272	. . . . . . 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 1040	. . . . 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 1550	1 |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  E!weu 1357  {cab 1440  Vcvv 1786   class class class wbr 2587  ` cfv 3145

This theorem is referenced by:  tz6.12-1 3675  tz6.12-2 3678

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-xp 3147  df-cnv 3149  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fv 3161

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