Theorem fnimapr 5737

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 5736	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )
2	1	3adant3 1044	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` B ) } = ( F " { B } ) )
3		fnsnfv 5736	. . . . 5 |- ( ( F Fn A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
4	3	3adant2 1043	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
5	2, 4	uneq12d 3374	. . 3 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( { ( F ` B ) } u. { ( F ` C ) } ) = ( ( F " { B } ) u. ( F " { C } ) ) )
6	5	eqcomd 2238	. 2 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( ( F " { B } ) u. ( F " { C } ) ) = ( { ( F ` B ) } u. { ( F ` C ) } ) )
7		df-pr 3696	. . . 4 |- { B , C } = ( { B } u. { C } )
8	7	imaeq2i 5099	. . 3 |- ( F " { B , C } ) = ( F " ( { B } u. { C } ) )
9		imaundi 5175	. . 3 |- ( F " ( { B } u. { C } ) ) = ( ( F " { B } ) u. ( F " { C } ) )
10	8, 9	eqtri 2253	. 2 |- ( F " { B , C } ) = ( ( F " { B } ) u. ( F " { C } ) )
11		df-pr 3696	. 2 |- { ( F ` B ) , ( F ` C ) } = ( { ( F ` B ) } u. { ( F ` C ) } )
12	6, 10, 11	3eqtr4g 2290	1 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   u. cun 3209   {csn 3689   {cpr 3690   "cima 4752   Fn wfn 5347   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  en2  7065  fvinim0ffz  10587