Theorem fnimapr 6262

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 6258	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )
2	1	3adant3 1081	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` B ) } = ( F " { B } ) )
3		fnsnfv 6258	. . . . 5 |- ( ( F Fn A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
4	3	3adant2 1080	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
5	2, 4	uneq12d 3768	. . 3 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( { ( F ` B ) } u. { ( F ` C ) } ) = ( ( F " { B } ) u. ( F " { C } ) ) )
6	5	eqcomd 2628	. 2 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( ( F " { B } ) u. ( F " { C } ) ) = ( { ( F ` B ) } u. { ( F ` C ) } ) )
7		df-pr 4180	. . . 4 |- { B , C } = ( { B } u. { C } )
8	7	imaeq2i 5464	. . 3 |- ( F " { B , C } ) = ( F " ( { B } u. { C } ) )
9		imaundi 5545	. . 3 |- ( F " ( { B } u. { C } ) ) = ( ( F " { B } ) u. ( F " { C } ) )
10	8, 9	eqtri 2644	. 2 |- ( F " { B , C } ) = ( ( F " { B } ) u. ( F " { C } ) )
11		df-pr 4180	. 2 |- { ( F ` B ) , ( F ` C ) } = ( { ( F ` B ) } u. { ( F ` C ) } )
12	6, 10, 11	3eqtr4g 2681	1 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   {csn 4177   {cpr 4179   "cima 5117   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fvinim0ffz  12587  mrcun  16282  poimirlem1  33410  poimirlem9  33418  imarnf1pr  41301