Theorem fv3 5341

Description: Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

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

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfv 5326	. . 3 |- (x e. (F` A) <-> E.z(x e. z /\ A.y(AFy <-> y = z)))
2		bi2 189	. . . . . . . . . 10 |- ((AFy <-> y = z) -> (y = z -> AFy))
3	2	alimi 1559	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> A.y(y = z -> AFy))
4		vex 2862	. . . . . . . . . 10 |- z e. _V
5		breq2 4643	. . . . . . . . . 10 |- (y = z -> (AFy <-> AFz))
6	4, 5	ceqsalv 2885	. . . . . . . . 9 |- (A.y(y = z -> AFy) <-> AFz)
7	3, 6	sylib 188	. . . . . . . 8 |- (A.y(AFy <-> y = z) -> AFz)
8	7	anim2i 552	. . . . . . 7 |- ((x e. z /\ A.y(AFy <-> y = z)) -> (x e. z /\ AFz))
9	8	eximi 1576	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.z(x e. z /\ AFz))
10		elequ2 1715	. . . . . . . 8 |- (z = y -> (x e. z <-> x e. y))
11		breq2 4643	. . . . . . . 8 |- (z = y -> (AFz <-> AFy))
12	10, 11	anbi12d 691	. . . . . . 7 |- (z = y -> ((x e. z /\ AFz) <-> (x e. y /\ AFy)))
13	12	cbvexv 2003	. . . . . 6 |- (E.z(x e. z /\ AFz) <-> E.y(x e. y /\ AFy))
14	9, 13	sylib 188	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.y(x e. y /\ AFy))
15		19.40 1609	. . . . . . 7 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.z x e. z /\ E.zA.y(AFy <-> y = z)))
16	15	simprd 449	. . . . . 6 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E.zA.y(AFy <-> y = z))
17		df-eu 2208	. . . . . 6 |- (E!y AFy <-> E.zA.y(AFy <-> y = z))
18	16, 17	sylibr 203	. . . . 5 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> E!y AFy)
19	14, 18	jca 518	. . . 4 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) -> (E.y(x e. y /\ AFy) /\ E!y AFy))
20		nfeu1 2214	. . . . . . 7 |- F/yE!y AFy
21		nfv 1619	. . . . . . . . 9 |- F/y x e. z
22		nfa1 1788	. . . . . . . . 9 |- F/yA.y(AFy <-> y = z)
23	21, 22	nfan 1824	. . . . . . . 8 |- F/y(x e. z /\ A.y(AFy <-> y = z))
24	23	nfex 1843	. . . . . . 7 |- F/yE.z(x e. z /\ A.y(AFy <-> y = z))
25	20, 24	nfim 1813	. . . . . 6 |- F/y(E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z)))
26		bi1 178	. . . . . . . . . . . . . 14 |- ((AFy <-> y = z) -> (AFy -> y = z))
27		ax-14 1714	. . . . . . . . . . . . . 14 |- (y = z -> (x e. y -> x e. z))
28	26, 27	syl6 29	. . . . . . . . . . . . 13 |- ((AFy <-> y = z) -> (AFy -> (x e. y -> x e. z)))
29	28	com23 72	. . . . . . . . . . . 12 |- ((AFy <-> y = z) -> (x e. y -> (AFy -> x e. z)))
30	29	imp3a 420	. . . . . . . . . . 11 |- ((AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
31	30	sps 1754	. . . . . . . . . 10 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> x e. z))
32	31	anc2ri 541	. . . . . . . . 9 |- (A.y(AFy <-> y = z) -> ((x e. y /\ AFy) -> (x e. z /\ A.y(AFy <-> y = z))))
33	32	com12 27	. . . . . . . 8 |- ((x e. y /\ AFy) -> (A.y(AFy <-> y = z) -> (x e. z /\ A.y(AFy <-> y = z))))
34	33	eximdv 1622	. . . . . . 7 |- ((x e. y /\ AFy) -> (E.zA.y(AFy <-> y = z) -> E.z(x e. z /\ A.y(AFy <-> y = z))))
35	17, 34	syl5bi 208	. . . . . 6 |- ((x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
36	25, 35	exlimi 1803	. . . . 5 |- (E.y(x e. y /\ AFy) -> (E!y AFy -> E.z(x e. z /\ A.y(AFy <-> y = z))))
37	36	imp 418	. . . 4 |- ((E.y(x e. y /\ AFy) /\ E!y AFy) -> E.z(x e. z /\ A.y(AFy <-> y = z)))
38	19, 37	impbii 180	. . 3 |- (E.z(x e. z /\ A.y(AFy <-> y = z)) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
39	1, 38	bitri 240	. 2 |- (x e. (F` A) <-> (E.y(x e. y /\ AFy) /\ E!y AFy))
40	39	abbi2i 2464	1 |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  {cab 2339   class class class wbr 4639  ` cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795

This theorem is referenced by:  tz6.12-2  5346