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 5174	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2791	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1547   ⊆ wss 3890   ↦ cmpt 5160   ↾ cres 5627  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  pwfseqlem5  10584  pfxres  14640  gsumpt  19935  dpjidcl  20033  gsumle  20118  regsumsupp  21604  tsmsxplem2  24144  dvmulbr  25931  dvlip  25985  lhop1lem  26005  loglesqrt  26750  jensenlem1  26975  jensen  26977  amgm  26979  ushgredgedg  29323  ushgredgedgloop  29325  fisuppov1  32782  fmptunsnop  32799  mgcf1o  33089  gsumfs2d  33149  gsumzresunsn  33150  gsumpart  33151  gsumhashmul  33155  rprmdvdsprod  33624  coinflippv  34675  fdvposlt  34790  fdvposle  34792  logdivsqrle  34841  ftc1cnnclem  38065  dvasin  38078  dvacos  38079  dvreasin  38080  dvreacos  38081  areacirclem1  38082  dvrelog2  42556  dvrelog3  42557  aks6d1c2  42622  aks6d1c6lem3  42664  readvrec2  42845  readvrec  42846  resuppsinopn  42847  cantnf2  43777  limsupvaluz2  46188  supcnvlimsup  46190  itgperiod  46431  fourierdlem69  46625  fourierdlem73  46629  fourierdlem74  46630  fourierdlem75  46631  fourierdlem76  46632  fourierdlem81  46637  fourierdlem85  46641  fourierdlem88  46644  fourierdlem92  46648  fourierdlem97  46653  fourierdlem100  46656  fourierdlem101  46657  fourierdlem103  46659  fourierdlem104  46660  fourierdlem107  46663  fourierdlem111  46667  fourierdlem112  46668  fouriersw  46681  sge0tsms  46830  sge0resrnlem  46853  meadjiunlem  46915  omeunle  46966  isomenndlem  46980