Theorem fnimapr 5652

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 5651	. . . . 5 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )
2	1	3adant3 1020	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` B ) } = ( F " { B } ) )
3		fnsnfv 5651	. . . . 5 |- ( ( F Fn A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
4	3	3adant2 1019	. . . 4 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> { ( F ` C ) } = ( F " { C } ) )
5	2, 4	uneq12d 3332	. . 3 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( { ( F ` B ) } u. { ( F ` C ) } ) = ( ( F " { B } ) u. ( F " { C } ) ) )
6	5	eqcomd 2212	. 2 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( ( F " { B } ) u. ( F " { C } ) ) = ( { ( F ` B ) } u. { ( F ` C ) } ) )
7		df-pr 3645	. . . 4 |- { B , C } = ( { B } u. { C } )
8	7	imaeq2i 5029	. . 3 |- ( F " { B , C } ) = ( F " ( { B } u. { C } ) )
9		imaundi 5104	. . 3 |- ( F " ( { B } u. { C } ) ) = ( ( F " { B } ) u. ( F " { C } ) )
10	8, 9	eqtri 2227	. 2 |- ( F " { B , C } ) = ( ( F " { B } ) u. ( F " { C } ) )
11		df-pr 3645	. 2 |- { ( F ` B ) , ( F ` C ) } = ( { ( F ` B ) } u. { ( F ` C ) } )
12	6, 10, 11	3eqtr4g 2264	1 |- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2177   u. cun 3168   {csn 3638   {cpr 3639   "cima 4686   Fn wfn 5275   ` cfv 5280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288

This theorem is referenced by:  en2  6926  fvinim0ffz  10392