Theorem tz6.12-2 3739

Description: Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27.

Assertion

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

Distinct variable groups:   y,A   y,F

Proof of Theorem tz6.12-2

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . . 6 |- (-. E!y AFy -> A.z -. E!y AFy)
2		eq0 2294	. . . . . . 7 |- ({x | E!y AFy} = (/) <-> A.z -. z e. {x | E!y AFy})
3		visset 1813	. . . . . . . . . 10 |- z e. V
4		pm4.2d 171	. . . . . . . . . 10 |- (x = z -> (E!y AFy <-> E!y AFy))
5	3, 4	elab 1897	. . . . . . . . 9 |- (z e. {x | E!y AFy} <-> E!y AFy)
6	5	negbii 187	. . . . . . . 8 |- (-. z e. {x | E!y AFy} <-> -. E!y AFy)
7	6	albii 999	. . . . . . 7 |- (A.z -. z e. {x | E!y AFy} <-> A.z -. E!y AFy)
8	2, 7	bitr2 174	. . . . . 6 |- (A.z -. E!y AFy <-> {x | E!y AFy} = (/))
9	1, 8	sylib 198	. . . . 5 |- (-. E!y AFy -> {x | E!y AFy} = (/))
10	9	sseq2d 2089	. . . 4 |- (-. E!y AFy -> ((F` A) (_ {x | E!y AFy} <-> (F` A) (_ (/)))
11		fveq2 3724	. . . . . 6 |- (z = A -> (F` z) = (F` A))
12		breq1 2622	. . . . . . . 8 |- (z = A -> (zFy <-> AFy))
13	12	eubidv 1386	. . . . . . 7 |- (z = A -> (E!y zFy <-> E!y AFy))
14	13	abbidv 1577	. . . . . 6 |- (z = A -> {x | E!y zFy} = {x | E!y AFy})
15	11, 14	sseq12d 2090	. . . . 5 |- (z = A -> ((F` z) (_ {x | E!y zFy} <-> (F` A) (_ {x | E!y AFy}))
16	3	fv3 3733	. . . . . 6 |- (F` z) = {x | (E.y(x e. y /\ zFy) /\ E!y zFy)}
17		pm3.27 323	. . . . . . 7 |- ((E.y(x e. y /\ zFy) /\ E!y zFy) -> E!y zFy)
18	17	ss2abi 2120	. . . . . 6 |- {x | (E.y(x e. y /\ zFy) /\ E!y zFy)} (_ {x | E!y zFy}
19	16, 18	eqsstr 2091	. . . . 5 |- (F` z) (_ {x | E!y zFy}
20	15, 19	vtoclg 1847	. . . 4 |- (A e. V -> (F` A) (_ {x | E!y AFy})
21	10, 20	syl5bi 208	. . 3 |- (-. E!y AFy -> (A e. V -> (F` A) (_ (/)))
22		ss0 2303	. . 3 |- ((F` A) (_ (/) -> (F` A) = (/))
23	21, 22	syl6com 53	. 2 |- (A e. V -> (-. E!y AFy -> (F` A) = (/)))
24		fvprc 3721	. . 3 |- (-. A e. V -> (F` A) = (/))
25	24	a1d 12	. 2 |- (-. A e. V -> (-. E!y AFy -> (F` A) = (/)))
26	23, 25	pm2.61i 126	1 |- (-. E!y AFy -> (F` A) = (/))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  {cab 1463  Vcvv 1811   (_ wss 2047  (/)c0 2280   class class class wbr 2619  ` cfv 3182

This theorem is referenced by:  tz6.12i 3741  ndmfv 3745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198

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