Theorem feqresmpt 6940

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 6735	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6939	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6890	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5199	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2816	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1563   ⊆ wss 3907   ↦ cmpt 5185   ↾ cres 5653  ⟶wf 6521  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533

This theorem is referenced by:  pwfseqlem5  10636  pfxres  14705  gsumpt  20020  dpjidcl  20118  gsumle  20203  regsumsupp  21729  tsmsxplem2  24268  dvmulbr  26055  dvlip  26109  lhop1lem  26129  loglesqrt  26880  jensenlem1  27105  jensen  27107  amgm  27109  ushgredgedg  29484  ushgredgedgloop  29486  fisuppov1  32936  fmptunsnop  32953  mgcf1o  33231  gsumfs2d  33289  gsumzresunsn  33290  gsumpart  33291  gsumhashmul  33295  rprmdvdsprod  33736  coinflippv  34786  fdvposlt  34898  fdvposle  34900  logdivsqrle  34949  ftc1cnnclem  38197  dvasin  38210  dvacos  38211  dvreasin  38212  dvreacos  38213  areacirclem1  38214  dvrelog2  42688  dvrelog3  42689  aks6d1c2  42754  aks6d1c6lem3  42796  readvrec2  42977  readvrec  42978  resuppsinopn  42979  cantnf2  43909  limsupvaluz2  46311  supcnvlimsup  46313  itgperiod  46554  fourierdlem69  46748  fourierdlem73  46752  fourierdlem74  46753  fourierdlem75  46754  fourierdlem76  46755  fourierdlem81  46760  fourierdlem85  46764  fourierdlem88  46767  fourierdlem92  46771  fourierdlem97  46776  fourierdlem100  46779  fourierdlem101  46780  fourierdlem103  46782  fourierdlem104  46783  fourierdlem107  46786  fourierdlem111  46790  fourierdlem112  46791  fouriersw  46804  sge0tsms  46953  sge0resrnlem  46976  meadjiunlem  47038  omeunle  47089  isomenndlem  47103