Theorem tz6.12-2 2845

Description: Theorem 6.12(2) of [TakeutiZaring] p. 27.

Assertion

Ref	Expression
tz6.12-2	|- (-. E!y AFy -> (F` A) = (/))

Distinct variable group(s):   y,A   y,F

Proof of Theorem tz6.12-2

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . . 7 |- (-. E!y AFy -> A.z -. E!y AFy)
2		eq0 1719	. . . . . . . 8 |- ({x | E!y AFy} = (/) <-> A.z -. z e. {x | E!y AFy})
3		visset 1350	. . . . . . . . . . 11 |- z e. V
4		pm4.2i 149	. . . . . . . . . . 11 |- (x = z -> (E!y AFy <-> E!y AFy))
5	3, 4	elab 1415	. . . . . . . . . 10 |- (z e. {x | E!y AFy} <-> E!y AFy)
6	5	negbii 162	. . . . . . . . 9 |- (-. z e. {x | E!y AFy} <-> -. E!y AFy)
7	6	bial 695	. . . . . . . 8 |- (A.z -. z e. {x | E!y AFy} <-> A.z -. E!y AFy)
8	2, 7	bitr2 152	. . . . . . 7 |- (A.z -. E!y AFy <-> {x | E!y AFy} = (/))
9	1, 8	sylib 173	. . . . . 6 |- (-. E!y AFy -> {x | E!y AFy} = (/))
10	9	sseq2d 1528	. . . . 5 |- (-. E!y AFy -> ((F` A) (_ {x | E!y AFy} <-> (F` A) (_ (/)))
11		fveq2 2832	. . . . . . 7 |- (z = A -> (F` z) = (F` A))
12		breq1 2065	. . . . . . . . 9 |- (z = A -> (zFy <-> AFy))
13	12	bieudv 1013	. . . . . . . 8 |- (z = A -> (E!y zFy <-> E!y AFy))
14	13	biabdv 1183	. . . . . . 7 |- (z = A -> {x | E!y zFy} = {x | E!y AFy})
15	11, 14	sseq12d 1529	. . . . . 6 |- (z = A -> ((F` z) (_ {x | E!y zFy} <-> (F` A) (_ {x | E!y AFy}))
16	3	fv3 2839	. . . . . . 7 |- (F` z) = {x | (E.y(x e. y /\ zFy) /\ E!y zFy)}
17		pm3.27 260	. . . . . . . 8 |- ((E.y(x e. y /\ zFy) /\ E!y zFy) -> E!y zFy)
18	17	ss2abi 1552	. . . . . . 7 |- {x | (E.y(x e. y /\ zFy) /\ E!y zFy)} (_ {x | E!y zFy}
19	16, 18	eqsstr 1530	. . . . . 6 |- (F` z) (_ {x | E!y zFy}
20	15, 19	vtoclg 1383	. . . . 5 |- (A e. V -> (F` A) (_ {x | E!y AFy})
21	10, 20	syl5bi 183	. . . 4 |- (-. E!y AFy -> (A e. V -> (F` A) (_ (/)))
22	21	com12 13	. . 3 |- (A e. V -> (-. E!y AFy -> (F` A) (_ (/)))
23		ss0 1727	. . 3 |- ((F` A) (_ (/) -> (F` A) = (/))
24	22, 23	syl6 23	. 2 |- (A e. V -> (-. E!y AFy -> (F` A) = (/)))
25		fvprc 2829	. . 3 |- (-. A e. V -> (F` A) = (/))
26	25	a1d 14	. 2 |- (-. A e. V -> (-. E!y AFy -> (F` A) = (/)))
27	24, 26	pm2.61i 110	1 |- (-. E!y AFy -> (F` A) = (/))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  E!weu 1007  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  (/)c0 1707   class class class wbr 2054  ` cfv 2422

This theorem is referenced by:  tz6.12i 2847  ndmfv 2848

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-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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