Theorem tz6.12-1 3736

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

Hypothesis

Ref	Expression
tz6.12.1	|- A e. V

Assertion

Ref	Expression
tz6.12-1	|- ((AFy /\ E!y AFy) -> (F` A) = y)

Distinct variable groups:   y,F   y,A

Proof of Theorem tz6.12-1

Step	Hyp	Ref	Expression
1		tz6.12.1	. . . . . . . 8 |- A e. V
2	1	fv3 3733	. . . . . . 7 |- (F` A) = {z | (E.y(z e. y /\ AFy) /\ E!y AFy)}
3	2	abeq2i 1570	. . . . . 6 |- (z e. (F` A) <-> (E.y(z e. y /\ AFy) /\ E!y AFy))
4		exancom 1054	. . . . . . . . 9 |- (E.y(z e. y /\ AFy) <-> E.y(AFy /\ z e. y))
5	4	anbi1i 481	. . . . . . . 8 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E.y(AFy /\ z e. y) /\ E!y AFy))
6		ancom 435	. . . . . . . 8 |- ((E.y(AFy /\ z e. y) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
7	5, 6	bitr 173	. . . . . . 7 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
8		eupick 1434	. . . . . . 7 |- ((E!y AFy /\ E.y(AFy /\ z e. y)) -> (AFy -> z e. y))
9	7, 8	sylbi 199	. . . . . 6 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) -> (AFy -> z e. y))
10	3, 9	sylbi 199	. . . . 5 |- (z e. (F` A) -> (AFy -> z e. y))
11	10	com12 11	. . . 4 |- (AFy -> (z e. (F` A) -> z e. y))
12	11	adantr 389	. . 3 |- ((AFy /\ E!y AFy) -> (z e. (F` A) -> z e. y))
13		19.8a 1029	. . . . . . 7 |- ((z e. y /\ AFy) -> E.y(z e. y /\ AFy))
14	13	anim1i 334	. . . . . 6 |- (((z e. y /\ AFy) /\ E!y AFy) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
15	14	anasss 440	. . . . 5 |- ((z e. y /\ (AFy /\ E!y AFy)) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
16	15, 3	sylibr 200	. . . 4 |- ((z e. y /\ (AFy /\ E!y AFy)) -> z e. (F` A))
17	16	expcom 374	. . 3 |- ((AFy /\ E!y AFy) -> (z e. y -> z e. (F` A)))
18	12, 17	impbid 516	. 2 |- ((AFy /\ E!y AFy) -> (z e. (F` A) <-> z e. y))
19	18	eqrdv 1473	1 |- ((AFy /\ E!y AFy) -> (F` A) = y)

Syntax hints:   -> wi 3   /\ wa 223   = 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:  tz6.12 3737  tz6.12c 3740  funbrfv 3750

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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