Theorem fnressn 6059

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 4026	. . . . . 6 |- ( x = B -> { x } = { B } )
2	1	reseq2d 5262	. . . . 5 |- ( x = B -> ( F |` { x } ) = ( F |` { B } ) )
3		fveq2 5848	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
4		opeq12 4205	. . . . . . 7 |- ( ( x = B /\ ( F ` x ) = ( F ` B ) ) -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
5	3, 4	mpdan 666	. . . . . 6 |- ( x = B -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
6	5	sneqd 4028	. . . . 5 |- ( x = B -> { <. x , ( F ` x ) >. } = { <. B , ( F ` B ) >. } )
7	2, 6	eqeq12d 2476	. . . 4 |- ( x = B -> ( ( F |` { x } ) = { <. x , ( F ` x ) >. } <-> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
8	7	imbi2d 314	. . 3 |- ( x = B -> ( ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) <-> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) ) )
9		vex 3109	. . . . . . 7 |- x e. _V
10	9	snss 4140	. . . . . 6 |- ( x e. A <-> { x } C_ A )
11		fnssres 5676	. . . . . 6 |- ( ( F Fn A /\ { x } C_ A ) -> ( F |` { x } ) Fn { x } )
12	10, 11	sylan2b 473	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) Fn { x } )
13		dffn2 5714	. . . . . 6 |- ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) : { x } --> _V )
14	9	fsn2 6047	. . . . . 6 |- ( ( F |` { x } ) : { x } --> _V <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
15		fvex 5858	. . . . . . . 8 |- ( ( F |` { x } ) ` x ) e. _V
16	15	biantrur 504	. . . . . . 7 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
17		ssnid 4045	. . . . . . . . . . 11 |- x e. { x }
18		fvres 5862	. . . . . . . . . . 11 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
19	17, 18	ax-mp 5	. . . . . . . . . 10 |- ( ( F |` { x } ) ` x ) = ( F ` x )
20	19	opeq2i 4207	. . . . . . . . 9 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
21	20	sneqi 4027	. . . . . . . 8 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
22	21	eqeq2i 2472	. . . . . . 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 433	. . 3 |- ( x e. A -> ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
27	8, 26	vtoclga 3170	. 2 |- ( B e. A -> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
28	27	impcom 428	1 |- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   C_ wss 3461   {csn 4016   <.cop 4022   |` cres 4990   Fn wfn 5565   -->wf 5566   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578

This theorem is referenced by:  funressn  6060  fressnfv  6061  fnsnsplit  6084  canthp1lem2  9020  fseq1p1m1  11756  dprd2da  17286  dmdprdpr  17293  dprdpr  17294  dpjlem  17295  pgpfaclem1  17327  islindf4  19040  xpstopnlem1  20476  ptcmpfi  20480  2pthlem1  24799  eupath2lem3  25181  ginvsn  25549  subfacp1lem5  28892  cvmliftlem10  29003