Theorem feqresmpt 6952

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

Hypotheses

Ref	Expression
feqmptd.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
feqresmpt.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
feqresmpt	⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

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

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

Proof of Theorem feqresmpt

Step	Hyp	Ref	Expression
1		feqmptd.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		feqresmpt.2	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3	1, 2	fssresd 6747	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6951	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6902	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5207	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2814	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1570   ⊆ wss 3906   ↦ cmpt 5193   ↾ cres 5665  ⟶wf 6534  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546

This theorem is referenced by:  pwfseqlem5  10649  pfxres  14719  gsumpt  20033  dpjidcl  20131  gsumle  20216  regsumsupp  21753  tsmsxplem2  24292  dvmulbr  26079  dvlip  26133  lhop1lem  26153  loglesqrt  26907  jensenlem1  27132  jensen  27134  amgm  27136  ushgredgedg  29560  ushgredgedgloop  29562  fisuppov1  33009  fmptunsnop  33026  mgcf1o  33304  gsumfs2d  33362  gsumzresunsn  33363  gsumpart  33364  gsumhashmul  33368  rprmdvdsprod  33805  coinflippv  34855  fdvposlt  34967  fdvposle  34969  logdivsqrle  35018  ftc1cnnclem  38323  dvasin  38336  dvacos  38337  dvreasin  38338  dvreacos  38339  areacirclem1  38340  dvrelog2  42812  dvrelog3  42813  aks6d1c2  42878  aks6d1c6lem3  42920  readvrec2  43103  readvrec  43104  resuppsinopn  43105  cantnf2  44035  limsupvaluz2  46435  supcnvlimsup  46437  itgperiod  46678  fourierdlem69  46872  fourierdlem73  46876  fourierdlem74  46877  fourierdlem75  46878  fourierdlem76  46879  fourierdlem81  46884  fourierdlem85  46888  fourierdlem88  46891  fourierdlem92  46895  fourierdlem97  46900  fourierdlem100  46903  fourierdlem101  46904  fourierdlem103  46906  fourierdlem104  46907  fourierdlem107  46910  fourierdlem111  46914  fourierdlem112  46915  fouriersw  46928  sge0tsms  47077  sge0resrnlem  47100  meadjiunlem  47162  omeunle  47213  isomenndlem  47227