Theorem tz6.12c 3725

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.

Hypothesis

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

Assertion

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

Distinct variable groups:   y,F   y,A

Proof of Theorem tz6.12c

Step	Hyp	Ref	Expression
1		breq2 2613	. . 3 |- ((F` A) = y -> (AF(F` A) <-> AFy))
2		euex 1387	. . . 4 |- (E!y AFy -> E.y AFy)
3		hbeu1 1381	. . . . . 6 |- (E!y AFy -> A.yE!y AFy)
4		ax-17 968	. . . . . 6 |- (AF(F` A) -> A.y AF(F` A))
5	3, 4	hbim 1004	. . . . 5 |- ((E!y AFy -> AF(F` A)) -> A.y(E!y AFy -> AF(F` A)))
6		tz6.12c.1	. . . . . . . . 9 |- A e. V
7	6	tz6.12-1 3721	. . . . . . . 8 |- ((AFy /\ E!y AFy) -> (F` A) = y)
8	7	expcom 374	. . . . . . 7 |- (E!y AFy -> (AFy -> (F` A) = y))
9	1	biimprd 154	. . . . . . 7 |- ((F` A) = y -> (AFy -> AF(F` A)))
10	8, 9	syli 54	. . . . . 6 |- (E!y AFy -> (AFy -> AF(F` A)))
11	10	com12 11	. . . . 5 |- (AFy -> (E!y AFy -> AF(F` A)))
12	5, 11	19.23ai 1060	. . . 4 |- (E.y AFy -> (E!y AFy -> AF(F` A)))
13	2, 12	mpcom 49	. . 3 |- (E!y AFy -> AF(F` A))
14	1, 13	syl5cbi 209	. 2 |- (E!y AFy -> ((F` A) = y -> AFy))
15	14, 8	impbid 514	1 |- (E!y AFy -> ((F` A) = y <-> AFy))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  Vcvv 1802   class class class wbr 2609  ` cfv 3172

This theorem is referenced by:  tz6.12i 3726  fnbrfvb 3738

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

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-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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