Theorem dfimafn2 5695

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 5694	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )
2		iunab 4017	. . 3 |- U_ x e. A { y | ( F ` x ) = y } = { y | E. x e. A ( F ` x ) = y }
3	1, 2	eqtr4di 2282	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { y | ( F ` x ) = y } )
4		df-sn 3675	. . . . 5 |- { ( F ` x ) } = { y | y = ( F ` x ) }
5		eqcom 2233	. . . . . 6 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
6	5	abbii 2347	. . . . 5 |- { y | y = ( F ` x ) } = { y | ( F ` x ) = y }
7	4, 6	eqtri 2252	. . . 4 |- { ( F ` x ) } = { y | ( F ` x ) = y }
8	7	a1i 9	. . 3 |- ( x e. A -> { ( F ` x ) } = { y | ( F ` x ) = y } )
9	8	iuneq2i 3988	. 2 |- U_ x e. A { ( F ` x ) } = U_ x e. A { y | ( F ` x ) = y }
10	3, 9	eqtr4di 2282	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { ( F ` x ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   C_ wss 3200   {csn 3669   U_ciun 3970   dom cdm 4725   "cima 4728   Fun wfun 5320   ` cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by: (None)