Theorem fressnfv 5382

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 3417	. . . . . 6 |- ( x = B -> { x } = { B } )
2		reseq2 4635	. . . . . . . 8 |- ( { x } = { B } -> ( F |` { x } ) = ( F |` { B } ) )
3	2	feq1d 5065	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { x } ) : { x } --> C <-> ( F |` { B } ) : { x } --> C ) )
4		feq2 5062	. . . . . . 7 |- ( { x } = { B } -> ( ( F |` { B } ) : { x } --> C <-> ( F |` { B } ) : { B } --> C ) )
5	3, 4	bitrd 186	. . . . . 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 5209	. . . . . 6 |- ( x = B -> ( F ` x ) = ( F ` B ) )
8	7	eleq1d 2148	. . . . 5 |- ( x = B -> ( ( F ` x ) e. C <-> ( F ` B ) e. C ) )
9	6, 8	bibi12d 233	. . . 4 |- ( x = B -> ( ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) <-> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) )
10	9	imbi2d 228	. . 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 5381	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
12		vsnid 3434	. . . . . . . . . 10 |- x e. { x }
13		fvres 5230	. . . . . . . . . 10 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
14	12, 13	ax-mp 7	. . . . . . . . 9 |- ( ( F |` { x } ) ` x ) = ( F ` x )
15	14	opeq2i 3582	. . . . . . . 8 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
16	15	sneqi 3418	. . . . . . 7 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
17	16	eqeq2i 2092	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
18		vex 2605	. . . . . . . 8 |- x e. _V
19	18	fsn2 5369	. . . . . . 7 |- ( ( F |` { x } ) : { x } --> C <-> ( ( ( F |` { x } ) ` x ) e. C /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
20	14	eleq1i 2145	. . . . . . . 8 |- ( ( ( F |` { x } ) ` x ) e. C <-> ( F ` x ) e. C )
21		iba 294	. . . . . . . 8 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } -> ( ( ( F |` { x } ) ` x ) e. C <-> ( ( ( F |` { x } ) ` x ) e. C /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) ) )
22	20, 21	syl5rbbr 193	. . . . . . 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	syl5bb 190	. . . . . 6 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) )
24	17, 23	sylbir 133	. . . . 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 114	. . 3 |- ( x e. A -> ( F Fn A -> ( ( F |` { x } ) : { x } --> C <-> ( F ` x ) e. C ) ) )
27	10, 26	vtoclga 2665	. 2 |- ( B e. A -> ( F Fn A -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) )
28	27	impcom 123	1 |- ( ( F Fn A /\ B e. A ) -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   {csn 3406   <.cop 3409   |` cres 4373   Fn wfn 4927   -->wf 4928   ` cfv 4932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940

This theorem is referenced by: (None)