Theorem feqresmpt 6820

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 6625	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6819	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6775	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5173	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2795	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3883   ↦ cmpt 5153   ↾ cres 5582  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426

This theorem is referenced by:  pwfseqlem5  10350  pfxres  14320  gsumpt  19478  dpjidcl  19576  regsumsupp  20739  tsmsxplem2  23213  dvmulbr  25008  dvlip  25062  lhop1lem  25082  loglesqrt  25816  jensenlem1  26041  jensen  26043  amgm  26045  ushgredgedg  27499  ushgredgedgloop  27501  mgcf1o  31183  gsumzresunsn  31216  gsumpart  31217  gsumhashmul  31218  gsumle  31252  coinflippv  32350  fdvposlt  32479  fdvposle  32481  logdivsqrle  32530  ftc1cnnclem  35775  dvasin  35788  dvacos  35789  dvreasin  35790  dvreacos  35791  areacirclem1  35792  dvrelog2  40000  dvrelog3  40001  limsupvaluz2  43169  supcnvlimsup  43171  itgperiod  43412  fourierdlem69  43606  fourierdlem73  43610  fourierdlem74  43611  fourierdlem75  43612  fourierdlem76  43613  fourierdlem81  43618  fourierdlem85  43622  fourierdlem88  43625  fourierdlem92  43629  fourierdlem97  43634  fourierdlem100  43637  fourierdlem101  43638  fourierdlem103  43640  fourierdlem104  43641  fourierdlem107  43644  fourierdlem111  43648  fourierdlem112  43649  fouriersw  43662  sge0tsms  43808  sge0resrnlem  43831  meadjiunlem  43893  omeunle  43944  isomenndlem  43958