Theorem feqresmpt 6891

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 6690	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6890	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6841	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5186	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2782	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3902   ↦ cmpt 5172   ↾ cres 5618  ⟶wf 6477  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489

This theorem is referenced by:  pwfseqlem5  10554  pfxres  14587  gsumpt  19875  dpjidcl  19973  gsumle  20058  regsumsupp  21560  tsmsxplem2  24070  dvmulbr  25869  dvmulbrOLD  25870  dvlip  25926  lhop1lem  25946  loglesqrt  26699  jensenlem1  26925  jensen  26927  amgm  26929  ushgredgedg  29208  ushgredgedgloop  29210  fisuppov1  32662  fmptunsnop  32679  mgcf1o  32982  gsumfs2d  33033  gsumzresunsn  33034  gsumpart  33035  gsumhashmul  33039  rprmdvdsprod  33497  coinflippv  34495  fdvposlt  34610  fdvposle  34612  logdivsqrle  34661  ftc1cnnclem  37737  dvasin  37750  dvacos  37751  dvreasin  37752  dvreacos  37753  areacirclem1  37754  dvrelog2  42103  dvrelog3  42104  aks6d1c2  42169  aks6d1c6lem3  42211  readvrec2  42400  readvrec  42401  resuppsinopn  42402  cantnf2  43364  limsupvaluz2  45782  supcnvlimsup  45784  itgperiod  46025  fourierdlem69  46219  fourierdlem73  46223  fourierdlem74  46224  fourierdlem75  46225  fourierdlem76  46226  fourierdlem81  46231  fourierdlem85  46235  fourierdlem88  46238  fourierdlem92  46242  fourierdlem97  46247  fourierdlem100  46250  fourierdlem101  46251  fourierdlem103  46253  fourierdlem104  46254  fourierdlem107  46257  fourierdlem111  46261  fourierdlem112  46262  fouriersw  46275  sge0tsms  46424  sge0resrnlem  46447  meadjiunlem  46509  omeunle  46560  isomenndlem  46574