Theorem dfimafn2 3768

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 (_ 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 3767	. . 3 |- ((Fun F /\ A (_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})
2		iunab 2601	. . 3 |- U_x e. A {y | (F` x) = y} = {y | E.x e. A (F` x) = y}
3	1, 2	syl6eqr 1528	. 2 |- ((Fun F /\ A (_ dom F) -> (F"A) = U_x e. A {y | (F` x) = y})
4		df-sn 2416	. . . . 5 |- {(F` x)} = {y | y = (F` x)}
5		eqcom 1480	. . . . . 6 |- (y = (F` x) <-> (F` x) = y)
6	5	abbii 1578	. . . . 5 |- {y | y = (F` x)} = {y | (F` x) = y}
7	4, 6	eqtr 1498	. . . 4 |- {(F` x)} = {y | (F` x) = y}
8	7	a1i 8	. . 3 |- (x e. A -> {(F` x)} = {y | (F` x) = y})
9	8	iuneq2i 2584	. 2 |- U_x e. A {(F` x)} = U_x e. A {y | (F` x) = y}
10	3, 9	syl6eqr 1528	1 |- ((Fun F /\ A (_ dom F) -> (F"A) = U_x e. A {(F` x)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  E.wrex 1649   (_ wss 2050  {csn 2413  U_ciun 2570  dom cdm 3176  "cima 3179  Fun wfun 3182  ` cfv 3188

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-9 967  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-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  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-iun 2572  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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