Theorem tz6.12i 4698

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 4689	. . . 4 |- (F` A) e. _V
2		eleq1 1957	. . . 4 |- ((F` A) = B -> ((F` A) e. _V <-> B e. _V))
3	1, 2	mpbii 210	. . 3 |- ((F` A) = B -> B e. _V)
4		eqeq2 1893	. . . . 5 |- (y = B -> ((F` A) = y <-> (F` A) = B))
5		neeq1 2024	. . . . . 6 |- (y = B -> (y =/= (/) <-> B =/= (/)))
6		breq2 3342	. . . . . 6 |- (y = B -> (AFy <-> AFB))
7	5, 6	imbi12d 688	. . . . 5 |- (y = B -> ((y =/= (/) -> AFy) <-> (B =/= (/) -> AFB)))
8	4, 7	imbi12d 688	. . . 4 |- (y = B -> (((F` A) = y -> (y =/= (/) -> AFy)) <-> ((F` A) = B -> (B =/= (/) -> AFB))))
9		neeq1 2024	. . . . . 6 |- ((F` A) = y -> ((F` A) =/= (/) <-> y =/= (/)))
10		tz6.12-2 4696	. . . . . . . . 9 |- (-. E!y AFy -> (F` A) = (/))
11	10	necon1ai 2047	. . . . . . . 8 |- ((F` A) =/= (/) -> E!y AFy)
12		tz6.12i.1	. . . . . . . . 9 |- A e. _V
13	12	tz6.12c 4697	. . . . . . . 8 |- (E!y AFy -> ((F` A) = y <-> AFy))
14	11, 13	syl 12	. . . . . . 7 |- ((F` A) =/= (/) -> ((F` A) = y <-> AFy))
15	14	biimpd 170	. . . . . 6 |- ((F` A) =/= (/) -> ((F` A) = y -> AFy))
16	9, 15	syl6bir 232	. . . . 5 |- ((F` A) = y -> (y =/= (/) -> ((F` A) = y -> AFy)))
17	16	pm2.43a 80	. . . 4 |- ((F` A) = y -> (y =/= (/) -> AFy))
18	8, 17	vtoclg 2346	. . 3 |- (B e. _V -> ((F` A) = B -> (B =/= (/) -> AFB)))
19	3, 18	mpcom 60	. 2 |- ((F` A) = B -> (B =/= (/) -> AFB))
20	19	com12 14	1 |- (B =/= (/) -> ((F` A) = B -> AFB))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  E!weu 1771   =/= wne 2017  _Vcvv 2292  (/)c0 2875   class class class wbr 3338  ` cfv 3998

This theorem is referenced by:  fvclss 4831

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014

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