Theorem dfimafn2 5464

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

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfimafn 5463	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )
2		iunab 3854	. . 3 |- U_ x e. A { y | ( F ` x ) = y } = { y | E. x e. A ( F ` x ) = y }
3	1, 2	syl6eqr 2188	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { y | ( F ` x ) = y } )
4		df-sn 3528	. . . . 5 |- { ( F ` x ) } = { y | y = ( F ` x ) }
5		eqcom 2139	. . . . . 6 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
6	5	abbii 2253	. . . . 5 |- { y | y = ( F ` x ) } = { y | ( F ` x ) = y }
7	4, 6	eqtri 2158	. . . 4 |- { ( F ` x ) } = { y | ( F ` x ) = y }
8	7	a1i 9	. . 3 |- ( x e. A -> { ( F ` x ) } = { y | ( F ` x ) = y } )
9	8	iuneq2i 3826	. 2 |- U_ x e. A { ( F ` x ) } = U_ x e. A { y | ( F ` x ) = y }
10	3, 9	syl6eqr 2188	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { ( F ` x ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2123   E.wrex 2415   C_ wss 3066   {csn 3522   U_ciun 3808   dom cdm 4534   "cima 4537   Fun wfun 5112   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by: (None)