Theorem tz6.12-1 4693

Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.

Hypothesis

Ref	Expression
tz6.12.1	|- A e. _V

Assertion

Ref	Expression
tz6.12-1	|- ((AFy /\ E!y AFy) -> (F` A) = y)

Distinct variable groups:   y,F   y,A

Proof of Theorem tz6.12-1

Step	Hyp	Ref	Expression
1		tz6.12.1	. . . . . . . 8 |- A e. _V
2	1	fv3 4690	. . . . . . 7 |- (F` A) = {z | (E.y(z e. y /\ AFy) /\ E!y AFy)}
3	2	abeq2i 2001	. . . . . 6 |- (z e. (F` A) <-> (E.y(z e. y /\ AFy) /\ E!y AFy))
4		exancom 1401	. . . . . . . . 9 |- (E.y(z e. y /\ AFy) <-> E.y(AFy /\ z e. y))
5	4	anbi1i 539	. . . . . . . 8 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E.y(AFy /\ z e. y) /\ E!y AFy))
6		ancom 482	. . . . . . . 8 |- ((E.y(AFy /\ z e. y) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
7	5, 6	bitri 190	. . . . . . 7 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
8		eupick 1834	. . . . . . 7 |- ((E!y AFy /\ E.y(AFy /\ z e. y)) -> (AFy -> z e. y))
9	7, 8	sylbi 216	. . . . . 6 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) -> (AFy -> z e. y))
10	3, 9	sylbi 216	. . . . 5 |- (z e. (F` A) -> (AFy -> z e. y))
11	10	com12 14	. . . 4 |- (AFy -> (z e. (F` A) -> z e. y))
12	11	adantr 425	. . 3 |- ((AFy /\ E!y AFy) -> (z e. (F` A) -> z e. y))
13		19.8a 1376	. . . . . . 7 |- ((z e. y /\ AFy) -> E.y(z e. y /\ AFy))
14	13	anim1i 361	. . . . . 6 |- (((z e. y /\ AFy) /\ E!y AFy) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
15	14	anasss 488	. . . . 5 |- ((z e. y /\ (AFy /\ E!y AFy)) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
16	15, 3	sylibr 217	. . . 4 |- ((z e. y /\ (AFy /\ E!y AFy)) -> z e. (F` A))
17	16	expcom 403	. . 3 |- ((AFy /\ E!y AFy) -> (z e. y -> z e. (F` A)))
18	12, 17	impbid 574	. 2 |- ((AFy /\ E!y AFy) -> (z e. (F` A) <-> z e. y))
19	18	eqrdv 1882	1 |- ((AFy /\ E!y AFy) -> (F` A) = y)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  _Vcvv 2292   class class class wbr 3338  ` cfv 3998

This theorem is referenced by:  tz6.12 4694  tz6.12c 4697  funbrfv 4709

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

Copyright terms: Public domain