Theorem dfimafn2 5746

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 5745	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )
2		iunab 4054	. . 3 |- U_ x e. A { y | ( F ` x ) = y } = { y | E. x e. A ( F ` x ) = y }
3	1, 2	eqtr4di 2289	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { y | ( F ` x ) = y } )
4		df-sn 3711	. . . . 5 |- { ( F ` x ) } = { y | y = ( F ` x ) }
5		eqcom 2240	. . . . . 6 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
6	5	abbii 2354	. . . . 5 |- { y | y = ( F ` x ) } = { y | ( F ` x ) = y }
7	4, 6	eqtri 2259	. . . 4 |- { ( F ` x ) } = { y | ( F ` x ) = y }
8	7	a1i 9	. . 3 |- ( x e. A -> { ( F ` x ) } = { y | ( F ` x ) = y } )
9	8	iuneq2i 4025	. 2 |- U_ x e. A { ( F ` x ) } = U_ x e. A { y | ( F ` x ) = y }
10	3, 9	eqtr4di 2289	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { ( F ` x ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   C_ wss 3220   {csn 3705   U_ciun 4007   dom cdm 4769   "cima 4772   Fun wfun 5366   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by: (None)