Theorem tz6.12-1 3727

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 3724	. . . . . . 7 |- (F` A) = {z | (E.y(z e. y /\ AFy) /\ E!y AFy)}
3	2	abeq2i 1567	. . . . . 6 |- (z e. (F` A) <-> (E.y(z e. y /\ AFy) /\ E!y AFy))
4		exancom 1052	. . . . . . . . 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 1432	. . . . . . 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 1027	. . . . . . 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 515	. 2 |- ((AFy /\ E!y AFy) -> (z e. (F` A) <-> z e. y))
19	18	eqrdv 1471	1 |- ((AFy /\ E!y AFy) -> (F` A) = y)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  E!weu 1378  Vcvv 1807   class class class wbr 2614  ` cfv 3177

This theorem is referenced by:  tz6.12 3728  tz6.12c 3731  funbrfv 3741

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193

Copyright terms: Public domain