Theorem feqresmpt 6933

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 6730	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6932	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6880	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5205	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2781	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3917   ↦ cmpt 5191   ↾ cres 5643  ⟶wf 6510  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522

This theorem is referenced by:  pwfseqlem5  10623  pfxres  14651  gsumpt  19899  dpjidcl  19997  regsumsupp  21538  tsmsxplem2  24048  dvmulbr  25848  dvmulbrOLD  25849  dvlip  25905  lhop1lem  25925  loglesqrt  26678  jensenlem1  26904  jensen  26906  amgm  26908  ushgredgedg  29163  ushgredgedgloop  29165  fisuppov1  32613  fmptunsnop  32630  mgcf1o  32936  gsumfs2d  33002  gsumzresunsn  33003  gsumpart  33004  gsumhashmul  33008  gsumle  33045  rprmdvdsprod  33512  coinflippv  34482  fdvposlt  34597  fdvposle  34599  logdivsqrle  34648  ftc1cnnclem  37692  dvasin  37705  dvacos  37706  dvreasin  37707  dvreacos  37708  areacirclem1  37709  dvrelog2  42059  dvrelog3  42060  aks6d1c2  42125  aks6d1c6lem3  42167  readvrec2  42356  readvrec  42357  resuppsinopn  42358  cantnf2  43321  limsupvaluz2  45743  supcnvlimsup  45745  itgperiod  45986  fourierdlem69  46180  fourierdlem73  46184  fourierdlem74  46185  fourierdlem75  46186  fourierdlem76  46187  fourierdlem81  46192  fourierdlem85  46196  fourierdlem88  46199  fourierdlem92  46203  fourierdlem97  46208  fourierdlem100  46211  fourierdlem101  46212  fourierdlem103  46214  fourierdlem104  46215  fourierdlem107  46218  fourierdlem111  46222  fourierdlem112  46223  fouriersw  46236  sge0tsms  46385  sge0resrnlem  46408  meadjiunlem  46470  omeunle  46521  isomenndlem  46535