Theorem feqresmpt 6759

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 6564	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6758	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6714	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5131	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2787	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1543   ⊆ wss 3853   ↦ cmpt 5120   ↾ cres 5538  ⟶wf 6354  ‘cfv 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366

This theorem is referenced by:  pwfseqlem5  10242  pfxres  14209  gsumpt  19301  dpjidcl  19399  regsumsupp  20538  tsmsxplem2  23005  dvmulbr  24790  dvlip  24844  lhop1lem  24864  loglesqrt  25598  jensenlem1  25823  jensen  25825  amgm  25827  ushgredgedg  27271  ushgredgedgloop  27273  mgcf1o  30954  gsumzresunsn  30987  gsumpart  30988  gsumhashmul  30989  gsumle  31023  coinflippv  32116  fdvposlt  32245  fdvposle  32247  logdivsqrle  32296  ftc1cnnclem  35534  dvasin  35547  dvacos  35548  dvreasin  35549  dvreacos  35550  areacirclem1  35551  dvrelog2  39754  dvrelog3  39755  limsupvaluz2  42897  supcnvlimsup  42899  itgperiod  43140  fourierdlem69  43334  fourierdlem73  43338  fourierdlem74  43339  fourierdlem75  43340  fourierdlem76  43341  fourierdlem81  43346  fourierdlem85  43350  fourierdlem88  43353  fourierdlem92  43357  fourierdlem97  43362  fourierdlem100  43365  fourierdlem101  43366  fourierdlem103  43368  fourierdlem104  43369  fourierdlem107  43372  fourierdlem111  43376  fourierdlem112  43377  fouriersw  43390  sge0tsms  43536  sge0resrnlem  43559  meadjiunlem  43621  omeunle  43672  isomenndlem  43686