Theorem fressnfv 5718

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 3617	. . . . . 6 |- ( x = B -> { x } = { B } )
2		reseq2 4916	. . . . . . . 8 |- ( { x } = { B } -> ( F |` { x } ) = ( F |` { B } ) )
3	2	feq1d 5366	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { x } --> C ) )
4		feq2 5363	. . . . . . 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 5529	. . . . . 6 |- ( x = B -> ( F ` x ) = ( F ` B ) )
8	7	eleq1d 2257	. . . . 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 5717	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
12		vsnid 3638	. . . . . . . . . 10 |- x e. { x }
13		fvres 5553	. . . . . . . . . 10 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
14	12, 13	ax-mp 5	. . . . . . . . 9 |- ( ( F |` { x } ) ` x ) = ( F ` x )
15	14	opeq2i 3796	. . . . . . . 8 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
16	15	sneqi 3618	. . . . . . 7 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
17	16	eqeq2i 2199	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
18		vex 2754	. . . . . . . 8 |- x e. _V
19	18	fsn2 5705	. . . . . . 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 2254	. . . . . . . 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 2817	. 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 1363   e. wcel 2159   {csn 3606   <.cop 3609   |` cres 4642   Fn wfn 5225   -->wf 5226   ` cfv 5230

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-reu 2474  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238

This theorem is referenced by: (None)