Theorem dfimafn2 5607

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 5606	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )
2		iunab 3960	. . 3 |- U_ x e. A { y | ( F ` x ) = y } = { y | E. x e. A ( F ` x ) = y }
3	1, 2	eqtr4di 2244	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { y | ( F ` x ) = y } )
4		df-sn 3625	. . . . 5 |- { ( F ` x ) } = { y | y = ( F ` x ) }
5		eqcom 2195	. . . . . 6 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
6	5	abbii 2309	. . . . 5 |- { y | y = ( F ` x ) } = { y | ( F ` x ) = y }
7	4, 6	eqtri 2214	. . . 4 |- { ( F ` x ) } = { y | ( F ` x ) = y }
8	7	a1i 9	. . 3 |- ( x e. A -> { ( F ` x ) } = { y | ( F ` x ) = y } )
9	8	iuneq2i 3931	. 2 |- U_ x e. A { ( F ` x ) } = U_ x e. A { y | ( F ` x ) = y }
10	3, 9	eqtr4di 2244	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { ( F ` x ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   E.wrex 2473   C_ wss 3154   {csn 3619   U_ciun 3913   dom cdm 4660   "cima 4663   Fun wfun 5249   ` cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by: (None)