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 5181	. 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 3890   ↦ cmpt 5167   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492

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 5231  ax-nul 5241  ax-pr 5370

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  pwfseqlem5  10577  pfxres  14633  gsumpt  19928  dpjidcl  20026  gsumle  20111  regsumsupp  21612  tsmsxplem2  24129  dvmulbr  25916  dvlip  25970  lhop1lem  25990  loglesqrt  26738  jensenlem1  26964  jensen  26966  amgm  26968  ushgredgedg  29312  ushgredgedgloop  29314  fisuppov1  32771  fmptunsnop  32788  mgcf1o  33078  gsumfs2d  33137  gsumzresunsn  33138  gsumpart  33139  gsumhashmul  33143  rprmdvdsprod  33609  coinflippv  34644  fdvposlt  34759  fdvposle  34761  logdivsqrle  34810  ftc1cnnclem  38026  dvasin  38039  dvacos  38040  dvreasin  38041  dvreacos  38042  areacirclem1  38043  dvrelog2  42517  dvrelog3  42518  aks6d1c2  42583  aks6d1c6lem3  42625  readvrec2  42807  readvrec  42808  resuppsinopn  42809  cantnf2  43771  limsupvaluz2  46184  supcnvlimsup  46186  itgperiod  46427  fourierdlem69  46621  fourierdlem73  46625  fourierdlem74  46626  fourierdlem75  46627  fourierdlem76  46628  fourierdlem81  46633  fourierdlem85  46637  fourierdlem88  46640  fourierdlem92  46644  fourierdlem97  46649  fourierdlem100  46652  fourierdlem101  46653  fourierdlem103  46655  fourierdlem104  46656  fourierdlem107  46659  fourierdlem111  46663  fourierdlem112  46664  fouriersw  46677  sge0tsms  46826  sge0resrnlem  46849  meadjiunlem  46911  omeunle  46962  isomenndlem  46976