Theorem dfimafn2 5367

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 5366	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )
2		iunab 3782	. . 3 |- U_ x e. A { y | ( F ` x ) = y } = { y | E. x e. A ( F ` x ) = y }
3	1, 2	syl6eqr 2139	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { y | ( F ` x ) = y } )
4		df-sn 3456	. . . . 5 |- { ( F ` x ) } = { y | y = ( F ` x ) }
5		eqcom 2091	. . . . . 6 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
6	5	abbii 2204	. . . . 5 |- { y | y = ( F ` x ) } = { y | ( F ` x ) = y }
7	4, 6	eqtri 2109	. . . 4 |- { ( F ` x ) } = { y | ( F ` x ) = y }
8	7	a1i 9	. . 3 |- ( x e. A -> { ( F ` x ) } = { y | ( F ` x ) = y } )
9	8	iuneq2i 3754	. 2 |- U_ x e. A { ( F ` x ) } = U_ x e. A { y | ( F ` x ) = y }
10	3, 9	syl6eqr 2139	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { ( F ` x ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   {cab 2075   E.wrex 2361   C_ wss 3000   {csn 3450   U_ciun 3736   dom cdm 4452   "cima 4455   Fun wfun 5022   ` cfv 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036

This theorem is referenced by: (None)