Theorem tz6.12i 3726

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 3717	. . . 4 |- (F` A) e. V
2		eleq1 1526	. . . 4 |- ((F` A) = B -> ((F` A) e. V <-> B e. V))
3	1, 2	mpbii 193	. . 3 |- ((F` A) = B -> B e. V)
4		eqeq2 1476	. . . . 5 |- (y = B -> ((F` A) = y <-> (F` A) = B))
5		neeq1 1582	. . . . . 6 |- (y = B -> (y =/= (/) <-> B =/= (/)))
6		breq2 2613	. . . . . 6 |- (y = B -> (AFy <-> AFB))
7	5, 6	imbi12d 624	. . . . 5 |- (y = B -> ((y =/= (/) -> AFy) <-> (B =/= (/) -> AFB)))
8	4, 7	imbi12d 624	. . . 4 |- (y = B -> (((F` A) = y -> (y =/= (/) -> AFy)) <-> ((F` A) = B -> (B =/= (/) -> AFB))))
9		neeq1 1582	. . . . . 6 |- ((F` A) = y -> ((F` A) =/= (/) <-> y =/= (/)))
10		tz6.12-2 3724	. . . . . . . . 9 |- (-. E!y AFy -> (F` A) = (/))
11	10	necon1ai 1600	. . . . . . . 8 |- ((F` A) =/= (/) -> E!y AFy)
12		tz6.12i.1	. . . . . . . . 9 |- A e. V
13	12	tz6.12c 3725	. . . . . . . 8 |- (E!y AFy -> ((F` A) = y <-> AFy))
14	11, 13	syl 10	. . . . . . 7 |- ((F` A) =/= (/) -> ((F` A) = y <-> AFy))
15	14	biimpd 153	. . . . . 6 |- ((F` A) =/= (/) -> ((F` A) = y -> AFy))
16	9, 15	syl6bir 215	. . . . 5 |- ((F` A) = y -> (y =/= (/) -> ((F` A) = y -> AFy)))
17	16	pm2.43a 66	. . . 4 |- ((F` A) = y -> (y =/= (/) -> AFy))
18	8, 17	vtoclg 1838	. . 3 |- (B e. V -> ((F` A) = B -> (B =/= (/) -> AFB)))
19	3, 18	mpcom 49	. 2 |- ((F` A) = B -> (B =/= (/) -> AFB))
20	19	com12 11	1 |- (B =/= (/) -> ((F` A) = B -> AFB))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  E!weu 1373   =/= wne 1577  Vcvv 1802  (/)c0 2270   class class class wbr 2609  ` cfv 3172

This theorem is referenced by:  fvclss 3840

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188

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