Theorem fressnfv 5893

Description: The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
fressnfv	|- ( ( F Fn A /\ B e. A ) -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) )

Proof of Theorem fressnfv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3716	. . . . . 6 |- ( x = B -> { x } = { B } )
2		reseq2 5053	. . . . . . . 8 |- ( { x } = { B } -> ( F |` { x } ) = ( F |` { B } ) )
3	2	feq1d 5515	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { x } --> C ) )
4		feq2 5512	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { B } ) : { x } --> C <-> ( F |` { B } ) : { B } --> C ) )
5	3, 4	bitrd 188	. . . . . 6 |- ( { x } = { B } -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { B } --> C ) )
6	1, 5	syl 14	. . . . 5 |- ( x = B -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { B } --> C ) )
7		fveq2 5690	. . . . . 6 |- ( x = B -> ( F ` x ) = ( F ` B ) )
8	7	eleq1d 2307	. . . . 5 |- ( x = B -> ( ( F ` x ) e. C <-> ( F ` B ) e. C ) )
9	6, 8	bibi12d 235	. . . 4 |- ( x = B -> ( ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) <-> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) )
10	9	imbi2d 230	. . 3 |- ( x = B -> ( ( F Fn A -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) ) <-> ( F Fn A -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) ) )
11		fnressn 5892	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
12		vsnid 3737	. . . . . . . . . 10 |- x e. { x }
13		fvres 5714	. . . . . . . . . 10 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
14	12, 13	ax-mp 5	. . . . . . . . 9 |- ( ( F |` { x } ) ` x ) = ( F ` x )
15	14	opeq2i 3903	. . . . . . . 8 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
16	15	sneqi 3717	. . . . . . 7 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
17	16	eqeq2i 2249	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
18		vex 2824	. . . . . . . 8 |- x e. _V
19	18	fsn2 5873	. . . . . . 7 |- ( ( F |` { x } ) : { x } --> C <-> ( ( ( F |` { x } ) ` x ) e. C /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
20		iba 300	. . . . . . . 8 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } -> ( ( ( F |` { x } ) ` x ) e. C <-> ( ( ( F |` { x } ) ` x ) e. C /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) ) )
21	14	eleq1i 2304	. . . . . . . 8 |- ( ( ( F |` { x } ) ` x ) e. C <-> ( F ` x ) e. C )
22	20, 21	bitr3di 195	. . . . . . 7 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } -> ( ( ( ( F |` { x } ) ` x ) e. C /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) <-> ( F ` x ) e. C ) )
23	19, 22	bitrid 192	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) )
24	17, 23	sylbir 135	. . . . 5 |- ( ( F |` { x } ) = { <. x , ( F ` x ) >. } -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) )
25	11, 24	syl 14	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) )
26	25	expcom 116	. . 3 |- ( x e. A -> ( F Fn A -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) ) )
27	10, 26	vtoclga 2889	. 2 |- ( B e. A -> ( F Fn A -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) )
28	27	impcom 125	1 |- ( ( F Fn A /\ B e. A ) -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {csn 3705   <.cop 3708   |` cres 4771   Fn wfn 5367   -->wf 5368   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by: (None)