Theorem fnimapr 5715

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 5714	. . . . 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 5714	. . . . 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 3364	. . 3 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( { ( F ` B ) } u. { ( F ` C ) } ) = ( ( F " { B } ) u. ( F " { C } ) ) )
6	5	eqcomd 2237	. 2 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( ( F " { B } ) u. ( F " { C } ) ) = ( { ( F ` B ) } u. { ( F ` C ) } ) )
7		df-pr 3680	. . . 4 |- { B , C } = ( { B } u. { C } )
8	7	imaeq2i 5080	. . 3 |- ( F " { B , C } ) = ( F " ( { B } u. { C } ) )
9		imaundi 5156	. . 3 |- ( F " ( { B } u. { C } ) ) = ( ( F " { B } ) u. ( F " { C } ) )
10	8, 9	eqtri 2252	. 2 |- ( F " { B , C } ) = ( ( F " { B } ) u. ( F " { C } ) )
11		df-pr 3680	. 2 |- { ( F ` B ) , ( F ` C ) } = ( { ( F ` B ) } u. { ( F ` C ) } )
12	6, 10, 11	3eqtr4g 2289	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 2202   u. cun 3199   {csn 3673   {cpr 3674   "cima 4734   Fn wfn 5328   ` cfv 5333

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  en2  7041  fvinim0ffz  10533