Theorem feqresmpt 6899

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 6697	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6898	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6849	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5190	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2784	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3898   ↦ cmpt 5176   ↾ cres 5623  ⟶wf 6484  ‘cfv 6488

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496

This theorem is referenced by:  pwfseqlem5  10563  pfxres  14591  gsumpt  19878  dpjidcl  19976  gsumle  20061  regsumsupp  21563  tsmsxplem2  24072  dvmulbr  25871  dvmulbrOLD  25872  dvlip  25928  lhop1lem  25948  loglesqrt  26701  jensenlem1  26927  jensen  26929  amgm  26931  ushgredgedg  29211  ushgredgedgloop  29213  fisuppov1  32670  fmptunsnop  32687  mgcf1o  32993  gsumfs2d  33044  gsumzresunsn  33045  gsumpart  33046  gsumhashmul  33050  rprmdvdsprod  33508  coinflippv  34520  fdvposlt  34635  fdvposle  34637  logdivsqrle  34686  ftc1cnnclem  37754  dvasin  37767  dvacos  37768  dvreasin  37769  dvreacos  37770  areacirclem1  37771  dvrelog2  42180  dvrelog3  42181  aks6d1c2  42246  aks6d1c6lem3  42288  readvrec2  42482  readvrec  42483  resuppsinopn  42484  cantnf2  43445  limsupvaluz2  45863  supcnvlimsup  45865  itgperiod  46106  fourierdlem69  46300  fourierdlem73  46304  fourierdlem74  46305  fourierdlem75  46306  fourierdlem76  46307  fourierdlem81  46312  fourierdlem85  46316  fourierdlem88  46319  fourierdlem92  46323  fourierdlem97  46328  fourierdlem100  46331  fourierdlem101  46332  fourierdlem103  46334  fourierdlem104  46335  fourierdlem107  46338  fourierdlem111  46342  fourierdlem112  46343  fouriersw  46356  sge0tsms  46505  sge0resrnlem  46528  meadjiunlem  46590  omeunle  46641  isomenndlem  46655