Theorem dfimafn2 4721

Description: Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
dfimafn2	|- ((Fun F /\ A C_ dom F) -> (F"A) = U_x e. A {(F` x)})

Distinct variable groups:   x,A   x,F

Proof of Theorem dfimafn2

Step	Hyp	Ref	Expression
1		dfimafn 4720	. . 3 |- ((Fun F /\ A C_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})
2		iunab 3300	. . 3 |- U_x e. A {y | (F` x) = y} = {y | E.x e. A (F` x) = y}
3	1, 2	syl6eqr 1946	. 2 |- ((Fun F /\ A C_ dom F) -> (F"A) = U_x e. A {y | (F` x) = y})
4		df-sn 3049	. . . . 5 |- {(F` x)} = {y | y = (F` x)}
5		eqcom 1886	. . . . . 6 |- (y = (F` x) <-> (F` x) = y)
6	5	abbii 2006	. . . . 5 |- {y | y = (F` x)} = {y | (F` x) = y}
7	4, 6	eqtri 1908	. . . 4 |- {(F` x)} = {y | (F` x) = y}
8	7	a1i 8	. . 3 |- (x e. A -> {(F` x)} = {y | (F` x) = y})
9	8	iuneq2i 3276	. 2 |- U_x e. A {(F` x)} = U_x e. A {y | (F` x) = y}
10	3, 9	syl6eqr 1946	1 |- ((Fun F /\ A C_ dom F) -> (F"A) = U_x e. A {(F` x)})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106   C_ wss 2593  {csn 3044  U_ciun 3255  dom cdm 3986  "cima 3989  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  imfstnrelc 14396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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