Theorem feqresmpt 6983

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 6777	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6982	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6933	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5204	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2814	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ↦ cmpt 5190   ↾ cres 5658  ⟶wf 6560  ‘cfv 6564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-fv 6572

This theorem is used by:  pwfseqlem5  10785  pfxres  14869  gsumpt  20218  dpjidcl  20316  gsumle  20401  regsumsupp  21972  tsmsxplem2  24517  dvmulbr  26303  dvlip  26357  lhop1lem  26377  loglesqrt  27135  jensenlem1  27360  jensen  27362  amgm  27364  ushgredgedg  29856  ushgredgedgloop  29858  fisuppov1  33363  fmptunsnop  33380  mgcf1o  33653  gsumfs2d  33711  gsumzresunsn  33712  gsumpart  33713  gsumhashmul  33717  zzedom  33952  rprmdvdsprod  34163  coinflippv  35220  fdvposlt  35332  fdvposle  35334  logdivsqrle  35383  ftc1cnnclem  38870  dvasin  38883  dvacos  38884  dvreasin  38885  dvreacos  38886  areacirclem1  38887  dvrelog2  43373  dvrelog3  43374  aks6d1c2  43439  aks6d1c6lem3  43481  readvrec2  43672  readvrec  43673  resuppsinopn  43674  cantnf2  44582  limsupvaluz2  47015  supcnvlimsup  47017  itgperiod  47258  fourierdlem69  47452  fourierdlem73  47456  fourierdlem74  47457  fourierdlem75  47458  fourierdlem76  47459  fourierdlem81  47464  fourierdlem85  47468  fourierdlem88  47471  fourierdlem92  47475  fourierdlem97  47480  fourierdlem100  47483  fourierdlem101  47484  fourierdlem103  47486  fourierdlem104  47487  fourierdlem107  47490  fourierdlem111  47494  fourierdlem112  47495  fouriersw  47508  sge0tsms  47657  sge0resrnlem  47680  meadjiunlem  47742  omeunle  47793  isomenndlem  47807