Theorem fnressn 6084

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

Assertion

Ref	Expression
fnressn	|- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )

Proof of Theorem fnressn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4043	. . . . . 6 |- ( x = B -> { x } = { B } )
2	1	reseq2d 5279	. . . . 5 |- ( x = B -> ( F |` { x } ) = ( F |` { B } ) )
3		fveq2 5872	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
4		opeq12 4221	. . . . . . 7 |- ( ( x = B /\ ( F ` x ) = ( F ` B ) ) -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
5	3, 4	mpdan 668	. . . . . 6 |- ( x = B -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
6	5	sneqd 4045	. . . . 5 |- ( x = B -> { <. x , ( F ` x ) >. } = { <. B , ( F ` B ) >. } )
7	2, 6	eqeq12d 2489	. . . 4 |- ( x = B -> ( ( F |` { x } ) = { <. x , ( F ` x ) >. } <-> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
8	7	imbi2d 316	. . 3 |- ( x = B -> ( ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) <-> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) ) )
9		vex 3121	. . . . . . 7 |- x e. _V
10	9	snss 4157	. . . . . 6 |- ( x e. A <-> { x } C_ A )
11		fnssres 5700	. . . . . 6 |- ( ( F Fn A /\ { x } C_ A ) -> ( F |` { x } ) Fn { x } )
12	10, 11	sylan2b 475	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) Fn { x } )
13		dffn2 5738	. . . . . 6 |- ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) : { x } --> _V )
14	9	fsn2 6072	. . . . . 6 |- ( ( F |` { x } ) : { x } --> _V <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
15		fvex 5882	. . . . . . . 8 |- ( ( F |` { x } ) ` x ) e. _V
16	15	biantrur 506	. . . . . . 7 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
17		ssnid 4062	. . . . . . . . . . 11 |- x e. { x }
18		fvres 5886	. . . . . . . . . . 11 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
19	17, 18	ax-mp 5	. . . . . . . . . 10 |- ( ( F |` { x } ) ` x ) = ( F ` x )
20	19	opeq2i 4223	. . . . . . . . 9 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
21	20	sneqi 4044	. . . . . . . 8 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
22	21	eqeq2i 2485	. . . . . . 7 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
23	16, 22	bitr3i 251	. . . . . 6 |- ( ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
24	13, 14, 23	3bitri 271	. . . . 5 |- ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
25	12, 24	sylib 196	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
26	25	expcom 435	. . 3 |- ( x e. A -> ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
27	8, 26	vtoclga 3182	. 2 |- ( B e. A -> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
28	27	impcom 430	1 |- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3118   C_ wss 3481   {csn 4033   <.cop 4039   |` cres 5007   Fn wfn 5589   -->wf 5590   ` cfv 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602

This theorem is referenced by:  funressn  6085  fressnfv  6086  fnsnsplit  6109  canthp1lem2  9043  fseq1p1m1  11764  dprd2da  16963  dmdprdpr  16970  dprdpr  16971  dpjlem  16972  pgpfaclem1  17004  islindf4  18742  xpstopnlem1  20178  ptcmpfi  20182  2pthlem1  24411  eupath2lem3  24793  ginvsn  25165  subfacp1lem5  28444  cvmliftlem10  28555