Theorem feqresmpt 6903

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 6701	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6902	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6853	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5193	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2787	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3901   ↦ cmpt 5179   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  pwfseqlem5  10574  pfxres  14603  gsumpt  19891  dpjidcl  19989  gsumle  20074  regsumsupp  21577  tsmsxplem2  24098  dvmulbr  25897  dvmulbrOLD  25898  dvlip  25954  lhop1lem  25974  loglesqrt  26727  jensenlem1  26953  jensen  26955  amgm  26957  ushgredgedg  29302  ushgredgedgloop  29304  fisuppov1  32762  fmptunsnop  32779  mgcf1o  33085  gsumfs2d  33144  gsumzresunsn  33145  gsumpart  33146  gsumhashmul  33150  rprmdvdsprod  33615  coinflippv  34641  fdvposlt  34756  fdvposle  34758  logdivsqrle  34807  ftc1cnnclem  37892  dvasin  37905  dvacos  37906  dvreasin  37907  dvreacos  37908  areacirclem1  37909  dvrelog2  42318  dvrelog3  42319  aks6d1c2  42384  aks6d1c6lem3  42426  readvrec2  42616  readvrec  42617  resuppsinopn  42618  cantnf2  43567  limsupvaluz2  45982  supcnvlimsup  45984  itgperiod  46225  fourierdlem69  46419  fourierdlem73  46423  fourierdlem74  46424  fourierdlem75  46425  fourierdlem76  46426  fourierdlem81  46431  fourierdlem85  46435  fourierdlem88  46438  fourierdlem92  46442  fourierdlem97  46447  fourierdlem100  46450  fourierdlem101  46451  fourierdlem103  46453  fourierdlem104  46454  fourierdlem107  46457  fourierdlem111  46461  fourierdlem112  46462  fouriersw  46475  sge0tsms  46624  sge0resrnlem  46647  meadjiunlem  46709  omeunle  46760  isomenndlem  46774