Theorem feqresmpt 6896

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 6695	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6895	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6845	. . 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 2780	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3905   ↦ cmpt 5176   ↾ cres 5625  ⟶wf 6482  ‘cfv 6486

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 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494

This theorem is referenced by:  pwfseqlem5  10576  pfxres  14604  gsumpt  19859  dpjidcl  19957  gsumle  20042  regsumsupp  21547  tsmsxplem2  24057  dvmulbr  25857  dvmulbrOLD  25858  dvlip  25914  lhop1lem  25934  loglesqrt  26687  jensenlem1  26913  jensen  26915  amgm  26917  ushgredgedg  29192  ushgredgedgloop  29194  fisuppov1  32639  fmptunsnop  32656  mgcf1o  32958  gsumfs2d  33021  gsumzresunsn  33022  gsumpart  33023  gsumhashmul  33027  rprmdvdsprod  33484  coinflippv  34454  fdvposlt  34569  fdvposle  34571  logdivsqrle  34620  ftc1cnnclem  37673  dvasin  37686  dvacos  37687  dvreasin  37688  dvreacos  37689  areacirclem1  37690  dvrelog2  42040  dvrelog3  42041  aks6d1c2  42106  aks6d1c6lem3  42148  readvrec2  42337  readvrec  42338  resuppsinopn  42339  cantnf2  43301  limsupvaluz2  45723  supcnvlimsup  45725  itgperiod  45966  fourierdlem69  46160  fourierdlem73  46164  fourierdlem74  46165  fourierdlem75  46166  fourierdlem76  46167  fourierdlem81  46172  fourierdlem85  46176  fourierdlem88  46179  fourierdlem92  46183  fourierdlem97  46188  fourierdlem100  46191  fourierdlem101  46192  fourierdlem103  46194  fourierdlem104  46195  fourierdlem107  46198  fourierdlem111  46202  fourierdlem112  46203  fouriersw  46216  sge0tsms  46365  sge0resrnlem  46388  meadjiunlem  46450  omeunle  46501  isomenndlem  46515