Theorem tz6.12-1 2842

Description: 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 group(s):   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 2839	. . . . . . 7 |- (F` A) = {z | (E.y(z e. y /\ AFy) /\ E!y AFy)}
3	2	cleqabi 1176	. . . . . 6 |- (z e. (F` A) <-> (E.y(z e. y /\ AFy) /\ E!y AFy))
4		exancom 736	. . . . . . . . 9 |- (E.y(z e. y /\ AFy) <-> E.y(AFy /\ z e. y))
5	4	anbi1i 368	. . . . . . . 8 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E.y(AFy /\ z e. y) /\ E!y AFy))
6		ancom 333	. . . . . . . 8 |- ((E.y(AFy /\ z e. y) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
7	5, 6	bitr 151	. . . . . . 7 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
8		eupick 1055	. . . . . . 7 |- ((E!y AFy /\ E.y(AFy /\ z e. y)) -> (AFy -> z e. y))
9	7, 8	sylbi 174	. . . . . 6 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) -> (AFy -> z e. y))
10	3, 9	sylbi 174	. . . . 5 |- (z e. (F` A) -> (AFy -> z e. y))
11	10	com12 13	. . . 4 |- (AFy -> (z e. (F` A) -> z e. y))
12	11	adantr 306	. . 3 |- ((AFy /\ E!y AFy) -> (z e. (F` A) -> z e. y))
13		19.8a 712	. . . . . . . 8 |- ((z e. y /\ AFy) -> E.y(z e. y /\ AFy))
14	13	anim1i 269	. . . . . . 7 |- (((z e. y /\ AFy) /\ E!y AFy) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
15	14	anasss 337	. . . . . 6 |- ((z e. y /\ (AFy /\ E!y AFy)) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
16	15, 3	sylibr 175	. . . . 5 |- ((z e. y /\ (AFy /\ E!y AFy)) -> z e. (F` A))
17	16	exp 291	. . . 4 |- (z e. y -> ((AFy /\ E!y AFy) -> z e. (F` A)))
18	17	com12 13	. . 3 |- ((AFy /\ E!y AFy) -> (z e. y -> z e. (F` A)))
19	12, 18	impbid 397	. 2 |- ((AFy /\ E!y AFy) -> (z e. (F` A) <-> z e. y))
20	19	cleqrd 1100	1 |- ((AFy /\ E!y AFy) -> (F` A) = y)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   e. wel 803  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  ` cfv 2422

This theorem is referenced by:  tz6.12 2843  tz6.12c 2846  funbrfv 2852  fvelrn 2883

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-mo 1010  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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