Theorem tz6.12i 3736

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 3727	. . . 4 |- (F` A) e. V
2		eleq1 1532	. . . 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 1482	. . . . 5 |- (y = B -> ((F` A) = y <-> (F` A) = B))
5		neeq1 1588	. . . . . 6 |- (y = B -> (y =/= (/) <-> B =/= (/)))
6		breq2 2619	. . . . . 6 |- (y = B -> (AFy <-> AFB))
7	5, 6	imbi12d 625	. . . . 5 |- (y = B -> ((y =/= (/) -> AFy) <-> (B =/= (/) -> AFB)))
8	4, 7	imbi12d 625	. . . 4 |- (y = B -> (((F` A) = y -> (y =/= (/) -> AFy)) <-> ((F` A) = B -> (B =/= (/) -> AFB))))
9		neeq1 1588	. . . . . 6 |- ((F` A) = y -> ((F` A) =/= (/) <-> y =/= (/)))
10		tz6.12-2 3734	. . . . . . . . 9 |- (-. E!y AFy -> (F` A) = (/))
11	10	necon1ai 1606	. . . . . . . 8 |- ((F` A) =/= (/) -> E!y AFy)
12		tz6.12i.1	. . . . . . . . 9 |- A e. V
13	12	tz6.12c 3735	. . . . . . . 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 1844	. . 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 955   e. wcel 957  E!weu 1379   =/= wne 1583  Vcvv 1808  (/)c0 2277   class class class wbr 2615  ` cfv 3178

This theorem is referenced by:  fvclss 3850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194

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