Theorem feqresmpt 6930

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 6725	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6929	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6880	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5192	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2812	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1559   ⊆ wss 3902   ↦ cmpt 5178   ↾ cres 5645  ⟶wf 6511  ‘cfv 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523

This theorem is referenced by:  pwfseqlem5  10614  pfxres  14686  gsumpt  19992  dpjidcl  20090  gsumle  20175  regsumsupp  21661  tsmsxplem2  24201  dvmulbr  25988  dvlip  26042  lhop1lem  26062  loglesqrt  26813  jensenlem1  27038  jensen  27040  amgm  27042  ushgredgedg  29386  ushgredgedgloop  29388  fisuppov1  32845  fmptunsnop  32862  mgcf1o  33141  gsumfs2d  33201  gsumzresunsn  33202  gsumpart  33203  gsumhashmul  33207  rprmdvdsprod  33690  coinflippv  34741  fdvposlt  34853  fdvposle  34855  logdivsqrle  34904  ftc1cnnclem  38150  dvasin  38163  dvacos  38164  dvreasin  38165  dvreacos  38166  areacirclem1  38167  dvrelog2  42641  dvrelog3  42642  aks6d1c2  42707  aks6d1c6lem3  42749  readvrec2  42930  readvrec  42931  resuppsinopn  42932  cantnf2  43862  limsupvaluz2  46272  supcnvlimsup  46274  itgperiod  46515  fourierdlem69  46709  fourierdlem73  46713  fourierdlem74  46714  fourierdlem75  46715  fourierdlem76  46716  fourierdlem81  46721  fourierdlem85  46725  fourierdlem88  46728  fourierdlem92  46732  fourierdlem97  46737  fourierdlem100  46740  fourierdlem101  46741  fourierdlem103  46743  fourierdlem104  46744  fourierdlem107  46747  fourierdlem111  46751  fourierdlem112  46752  fouriersw  46765  sge0tsms  46914  sge0resrnlem  46937  meadjiunlem  46999  omeunle  47050  isomenndlem  47064