Theorem fressnfv 5871

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 3700	. . . . . 6 |- ( x = B -> { x } = { B } )
2		reseq2 5033	. . . . . . . 8 |- ( { x } = { B } -> ( F |` { x } ) = ( F |` { B } ) )
3	2	feq1d 5495	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { x } --> C ) )
4		feq2 5492	. . . . . . 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 5670	. . . . . 6 |- ( x = B -> ( F ` x ) = ( F ` B ) )
8	7	eleq1d 2301	. . . . 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 5870	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
12		vsnid 3721	. . . . . . . . . 10 |- x e. { x }
13		fvres 5694	. . . . . . . . . 10 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
14	12, 13	ax-mp 5	. . . . . . . . 9 |- ( ( F |` { x } ) ` x ) = ( F ` x )
15	14	opeq2i 3887	. . . . . . . 8 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
16	15	sneqi 3701	. . . . . . 7 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
17	16	eqeq2i 2243	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
18		vex 2816	. . . . . . . 8 |- x e. _V
19	18	fsn2 5851	. . . . . . 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 2298	. . . . . . . 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 2881	. 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 1398   e. wcel 2203   {csn 3689   <.cop 3692   |` cres 4751   Fn wfn 5347   -->wf 5348   ` 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-reu 2527  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-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by: (None)