Theorem fv2 3720

Description: Alternate definition of function value. Definition 10.11 of [Quine] p. 68.

Hypothesis

Ref	Expression
fv2.1	|- A e. V

Assertion

Ref	Expression
fv2	|- (F` A) = U.{x | A.y(AFy <-> y = x)}

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem fv2

Step	Hyp	Ref	Expression
1		df-fv 3198	. 2 |- (F` A) = U.{x | (F"{A}) = {x}}
2		dfcleq 1470	. . . . 5 |- ((F"{A}) = {x} <-> A.y(y e. (F"{A}) <-> y e. {x}))
3		dfima2 3405	. . . . . . . . . 10 |- (F"{A}) = {y | E.x e. {A}xFy}
4	3	abeq2i 1570	. . . . . . . . 9 |- (y e. (F"{A}) <-> E.x e. {A}xFy)
5		df-rex 1650	. . . . . . . . 9 |- (E.x e. {A}xFy <-> E.x(x e. {A} /\ xFy))
6	4, 5	bitr 173	. . . . . . . 8 |- (y e. (F"{A}) <-> E.x(x e. {A} /\ xFy))
7		elsn 2421	. . . . . . . . . 10 |- (x e. {A} <-> x = A)
8	7	anbi1i 481	. . . . . . . . 9 |- ((x e. {A} /\ xFy) <-> (x = A /\ xFy))
9	8	exbii 1051	. . . . . . . 8 |- (E.x(x e. {A} /\ xFy) <-> E.x(x = A /\ xFy))
10		fv2.1	. . . . . . . . 9 |- A e. V
11		breq1 2622	. . . . . . . . 9 |- (x = A -> (xFy <-> AFy))
12	10, 11	ceqsexv 1835	. . . . . . . 8 |- (E.x(x = A /\ xFy) <-> AFy)
13	6, 9, 12	3bitr 177	. . . . . . 7 |- (y e. (F"{A}) <-> AFy)
14		elsn 2421	. . . . . . 7 |- (y e. {x} <-> y = x)
15	13, 14	bibi12i 610	. . . . . 6 |- ((y e. (F"{A}) <-> y e. {x}) <-> (AFy <-> y = x))
16	15	albii 999	. . . . 5 |- (A.y(y e. (F"{A}) <-> y e. {x}) <-> A.y(AFy <-> y = x))
17	2, 16	bitr 173	. . . 4 |- ((F"{A}) = {x} <-> A.y(AFy <-> y = x))
18	17	abbii 1575	. . 3 |- {x | (F"{A}) = {x}} = {x | A.y(AFy <-> y = x)}
19	18	unieqi 2511	. 2 |- U.{x | (F"{A}) = {x}} = U.{x | A.y(AFy <-> y = x)}
20	1, 19	eqtr 1495	1 |- (F` A) = U.{x | A.y(AFy <-> y = x)}

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  Vcvv 1811  {csn 2409  U.cuni 2503   class class class wbr 2619  "cima 3173  ` cfv 3182

This theorem is referenced by:  elfv 3722

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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