Theorem tz6.12c 3740

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 2623	. . 3 |- ((F` A) = y -> (AF(F` A) <-> AFy))
2		euex 1394	. . . 4 |- (E!y AFy -> E.y AFy)
3		hbeu1 1388	. . . . . 6 |- (E!y AFy -> A.yE!y AFy)
4		ax-17 971	. . . . . 6 |- (AF(F` A) -> A.y AF(F` A))
5	3, 4	hbim 1007	. . . . 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 3736	. . . . . . . 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 1064	. . . 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 516	1 |- (E!y AFy -> ((F` A) = y <-> AFy))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  Vcvv 1811   class class class wbr 2619  ` cfv 3182

This theorem is referenced by:  tz6.12i 3741  fnbrfvb 3753

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198

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