Theorem tz6.12c 3746

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 2628	. . 3 |- ((F` A) = y -> (AF(F` A) <-> AFy))
2		euex 1396	. . . 4 |- (E!y AFy -> E.y AFy)
3		hbeu1 1390	. . . . . 6 |- (E!y AFy -> A.yE!y AFy)
4		ax-17 973	. . . . . 6 |- (AF(F` A) -> A.y AF(F` A))
5	3, 4	hbim 1009	. . . . 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 3742	. . . . . . . 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 1066	. . . 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 518	1 |- (E!y AFy -> ((F` A) = y <-> AFy))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  E.wex 982  E!weu 1382  Vcvv 1814   class class class wbr 2624  ` cfv 3188

This theorem is referenced by:  tz6.12i 3747  fnbrfvb 3759

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204

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