Theorem feqresmpt 6911

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 6709	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6910	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6861	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5195	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2788	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3903   ↦ cmpt 5181   ↾ cres 5634  ⟶wf 6496  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508

This theorem is referenced by:  pwfseqlem5  10586  pfxres  14615  gsumpt  19903  dpjidcl  20001  gsumle  20086  regsumsupp  21589  tsmsxplem2  24110  dvmulbr  25909  dvmulbrOLD  25910  dvlip  25966  lhop1lem  25986  loglesqrt  26739  jensenlem1  26965  jensen  26967  amgm  26969  ushgredgedg  29314  ushgredgedgloop  29316  fisuppov1  32772  fmptunsnop  32789  mgcf1o  33095  gsumfs2d  33154  gsumzresunsn  33155  gsumpart  33156  gsumhashmul  33160  rprmdvdsprod  33626  coinflippv  34661  fdvposlt  34776  fdvposle  34778  logdivsqrle  34827  ftc1cnnclem  37939  dvasin  37952  dvacos  37953  dvreasin  37954  dvreacos  37955  areacirclem1  37956  dvrelog2  42431  dvrelog3  42432  aks6d1c2  42497  aks6d1c6lem3  42539  readvrec2  42728  readvrec  42729  resuppsinopn  42730  cantnf2  43679  limsupvaluz2  46093  supcnvlimsup  46095  itgperiod  46336  fourierdlem69  46530  fourierdlem73  46534  fourierdlem74  46535  fourierdlem75  46536  fourierdlem76  46537  fourierdlem81  46542  fourierdlem85  46546  fourierdlem88  46549  fourierdlem92  46553  fourierdlem97  46558  fourierdlem100  46561  fourierdlem101  46562  fourierdlem103  46564  fourierdlem104  46565  fourierdlem107  46568  fourierdlem111  46572  fourierdlem112  46573  fouriersw  46586  sge0tsms  46735  sge0resrnlem  46758  meadjiunlem  46820  omeunle  46871  isomenndlem  46885