Theorem feqresmpt 6250

Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Hypotheses

Ref	Expression
feqmptd.1	|- ( ph -> F : A --> B )
feqresmpt.2	|- ( ph -> C C_ A )

Assertion

Ref	Expression
feqresmpt	|- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Distinct variable groups:   x, A   x, C   x, F

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem feqresmpt

Step	Hyp	Ref	Expression
1		feqmptd.1	. . . 4 |- ( ph -> F : A --> B )
2		feqresmpt.2	. . . 4 |- ( ph -> C C_ A )
3	1, 2	fssresd 6071	. . 3 |- ( ph -> ( F |` C ) : C --> B )
4	3	feqmptd 6249	. 2 |- ( ph -> ( F |` C ) = ( x e. C |-> ( ( F |` C ) ` x ) ) )
5		fvres 6207	. . 3 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
6	5	mpteq2ia 4740	. 2 |- ( x e. C |-> ( ( F |` C ) ` x ) ) = ( x e. C |-> ( F ` x ) )
7	4, 6	syl6eq 2672	1 |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   |-> cmpt 4729   |` cres 5116   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  pwfseqlem5  9485  swrd0val  13421  gsumpt  18361  dpjidcl  18457  regsumsupp  19968  tsmsxplem2  21957  dvmulbr  23702  dvlip  23756  lhop1lem  23776  loglesqrt  24499  jensenlem1  24713  jensen  24715  amgm  24717  ushgredgedg  26121  ushgredgedgloop  26123  gsumle  29779  coinflippv  30545  fdvposlt  30677  fdvposle  30679  logdivsqrle  30728  ftc1cnnclem  33483  dvasin  33496  dvacos  33497  dvreasin  33498  dvreacos  33499  areacirclem1  33500  limsupvaluz2  39970  supcnvlimsup  39972  itgperiod  40197  fourierdlem69  40392  fourierdlem73  40396  fourierdlem74  40397  fourierdlem75  40398  fourierdlem76  40399  fourierdlem81  40404  fourierdlem85  40408  fourierdlem88  40411  fourierdlem92  40415  fourierdlem97  40420  fourierdlem100  40423  fourierdlem101  40424  fourierdlem103  40426  fourierdlem104  40427  fourierdlem107  40430  fourierdlem111  40434  fourierdlem112  40435  fouriersw  40448  sge0tsms  40597  sge0resrnlem  40620  meadjiunlem  40682  omeunle  40730  isomenndlem  40744  pfxres  41388