Theorem tz6.12i 2847

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

Hypothesis

Ref	Expression
tz6.12i.1	|- A e. V

Assertion

Ref	Expression
tz6.12i	|- (-. B = (/) -> ((F` A) = B -> AFB))

Proof of Theorem tz6.12i

Step	Hyp	Ref	Expression
1		fvex 2838	. . . 4 |- (F` A) e. V
2		eleq1 1149	. . . 4 |- ((F` A) = B -> ((F` A) e. V <-> B e. V))
3	1, 2	mpbii 168	. . 3 |- ((F` A) = B -> B e. V)
4		cleq2 1110	. . . . 5 |- (y = B -> ((F` A) = y <-> (F` A) = B))
5		cleq1 1107	. . . . . . 7 |- (y = B -> (y = (/) <-> B = (/)))
6	5	negbid 463	. . . . . 6 |- (y = B -> (-. y = (/) <-> -. B = (/)))
7		breq2 2066	. . . . . 6 |- (y = B -> (AFy <-> AFB))
8	6, 7	imbi12d 474	. . . . 5 |- (y = B -> ((-. y = (/) -> AFy) <-> (-. B = (/) -> AFB)))
9	4, 8	imbi12d 474	. . . 4 |- (y = B -> (((F` A) = y -> (-. y = (/) -> AFy)) <-> ((F` A) = B -> (-. B = (/) -> AFB))))
10		cleq1 1107	. . . . . . 7 |- ((F` A) = y -> ((F` A) = (/) <-> y = (/)))
11	10	negbid 463	. . . . . 6 |- ((F` A) = y -> (-. (F` A) = (/) <-> -. y = (/)))
12		tz6.12i.1	. . . . . . . . 9 |- A e. V
13	12	tz6.12c 2846	. . . . . . . 8 |- (E!y AFy -> ((F` A) = y <-> AFy))
14		tz6.12-2 2845	. . . . . . . 8 |- (-. E!y AFy -> (F` A) = (/))
15	13, 14	nsyl4 105	. . . . . . 7 |- (-. (F` A) = (/) -> ((F` A) = y <-> AFy))
16	15	biimpd 135	. . . . . 6 |- (-. (F` A) = (/) -> ((F` A) = y -> AFy))
17	11, 16	syl6bir 188	. . . . 5 |- ((F` A) = y -> (-. y = (/) -> ((F` A) = y -> AFy)))
18	17	pm2.43a 60	. . . 4 |- ((F` A) = y -> (-. y = (/) -> AFy))
19	9, 18	vtoclg 1383	. . 3 |- (B e. V -> ((F` A) = B -> (-. B = (/) -> AFB)))
20	3, 19	mpcom 49	. 2 |- ((F` A) = B -> (-. B = (/) -> AFB))
21	20	com12 13	1 |- (-. B = (/) -> ((F` A) = B -> AFB))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707   class class class wbr 2054  ` cfv 2422

This theorem is referenced by:  fvclss 2907

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-un 1076  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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