Theorem fnimapr 5618

Description: The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)

Assertion

Ref	Expression
fnimapr	|- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } )

Proof of Theorem fnimapr

Step	Hyp	Ref	Expression
1		fnsnfv 5617	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )
2	1	3adant3 1019	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` B ) } = ( F " { B } ) )
3		fnsnfv 5617	. . . . 5 |- ( ( F Fn A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
4	3	3adant2 1018	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
5	2, 4	uneq12d 3315	. . 3 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( { ( F ` B ) } u. { ( F ` C ) } ) = ( ( F " { B } ) u. ( F " { C } ) ) )
6	5	eqcomd 2199	. 2 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( ( F " { B } ) u. ( F " { C } ) ) = ( { ( F ` B ) } u. { ( F ` C ) } ) )
7		df-pr 3626	. . . 4 |- { B , C } = ( { B } u. { C } )
8	7	imaeq2i 5004	. . 3 |- ( F " { B , C } ) = ( F " ( { B } u. { C } ) )
9		imaundi 5079	. . 3 |- ( F " ( { B } u. { C } ) ) = ( ( F " { B } ) u. ( F " { C } ) )
10	8, 9	eqtri 2214	. 2 |- ( F " { B , C } ) = ( ( F " { B } ) u. ( F " { C } ) )
11		df-pr 3626	. 2 |- { ( F ` B ) , ( F ` C ) } = ( { ( F ` B ) } u. { ( F ` C ) } )
12	6, 10, 11	3eqtr4g 2251	1 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2164   u. cun 3152   {csn 3619   {cpr 3620   "cima 4663   Fn wfn 5250   ` 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-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:  fvinim0ffz  10311