Theorem feqresmpt 6967

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 6764	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6966	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6915	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5252	. 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 1533   ⊆ wss 3944   ↦ cmpt 5232   ↾ cres 5680  ⟶wf 6545  ‘cfv 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557

This theorem is referenced by:  pwfseqlem5  10688  pfxres  14665  gsumpt  19929  dpjidcl  20027  regsumsupp  21571  tsmsxplem2  24102  dvmulbr  25913  dvmulbrOLD  25914  dvlip  25970  lhop1lem  25990  loglesqrt  26738  jensenlem1  26964  jensen  26966  amgm  26968  ushgredgedg  29114  ushgredgedgloop  29116  mgcf1o  32819  gsumzresunsn  32858  gsumpart  32859  gsumhashmul  32860  gsumle  32894  rprmdvdsprod  33346  coinflippv  34234  fdvposlt  34362  fdvposle  34364  logdivsqrle  34413  ftc1cnnclem  37295  dvasin  37308  dvacos  37309  dvreasin  37310  dvreacos  37311  areacirclem1  37312  dvrelog2  41667  dvrelog3  41668  aks6d1c2  41733  aks6d1c6lem3  41775  cantnf2  42896  limsupvaluz2  45264  supcnvlimsup  45266  itgperiod  45507  fourierdlem69  45701  fourierdlem73  45705  fourierdlem74  45706  fourierdlem75  45707  fourierdlem76  45708  fourierdlem81  45713  fourierdlem85  45717  fourierdlem88  45720  fourierdlem92  45724  fourierdlem97  45729  fourierdlem100  45732  fourierdlem101  45733  fourierdlem103  45735  fourierdlem104  45736  fourierdlem107  45739  fourierdlem111  45743  fourierdlem112  45744  fouriersw  45757  sge0tsms  45906  sge0resrnlem  45929  meadjiunlem  45991  omeunle  46042  isomenndlem  46056