Theorem fressnfv 5749

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 3633	. . . . . 6 |- ( x = B -> { x } = { B } )
2		reseq2 4941	. . . . . . . 8 |- ( { x } = { B } -> ( F |` { x } ) = ( F |` { B } ) )
3	2	feq1d 5394	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { x } --> C ) )
4		feq2 5391	. . . . . . 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 5558	. . . . . 6 |- ( x = B -> ( F ` x ) = ( F ` B ) )
8	7	eleq1d 2265	. . . . 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 5748	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
12		vsnid 3654	. . . . . . . . . 10 |- x e. { x }
13		fvres 5582	. . . . . . . . . 10 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
14	12, 13	ax-mp 5	. . . . . . . . 9 |- ( ( F |` { x } ) ` x ) = ( F ` x )
15	14	opeq2i 3812	. . . . . . . 8 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
16	15	sneqi 3634	. . . . . . 7 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
17	16	eqeq2i 2207	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
18		vex 2766	. . . . . . . 8 |- x e. _V
19	18	fsn2 5736	. . . . . . 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 2262	. . . . . . . 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 2830	. 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 1364   e. wcel 2167   {csn 3622   <.cop 3625   |` cres 4665   Fn wfn 5253   -->wf 5254   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266

This theorem is referenced by: (None)