Theorem feqresmpt 6991

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 6788	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6990	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6939	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5269	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2796	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1537   ⊆ wss 3976   ↦ cmpt 5249   ↾ cres 5702  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  pwfseqlem5  10732  pfxres  14727  gsumpt  20004  dpjidcl  20102  regsumsupp  21663  tsmsxplem2  24183  dvmulbr  25995  dvmulbrOLD  25996  dvlip  26052  lhop1lem  26072  loglesqrt  26822  jensenlem1  27048  jensen  27050  amgm  27052  ushgredgedg  29264  ushgredgedgloop  29266  mgcf1o  32976  gsumzresunsn  33037  gsumpart  33038  gsumhashmul  33040  gsumle  33074  rprmdvdsprod  33527  coinflippv  34448  fdvposlt  34576  fdvposle  34578  logdivsqrle  34627  ftc1cnnclem  37651  dvasin  37664  dvacos  37665  dvreasin  37666  dvreacos  37667  areacirclem1  37668  dvrelog2  42021  dvrelog3  42022  aks6d1c2  42087  aks6d1c6lem3  42129  cantnf2  43287  limsupvaluz2  45659  supcnvlimsup  45661  itgperiod  45902  fourierdlem69  46096  fourierdlem73  46100  fourierdlem74  46101  fourierdlem75  46102  fourierdlem76  46103  fourierdlem81  46108  fourierdlem85  46112  fourierdlem88  46115  fourierdlem92  46119  fourierdlem97  46124  fourierdlem100  46127  fourierdlem101  46128  fourierdlem103  46130  fourierdlem104  46131  fourierdlem107  46134  fourierdlem111  46138  fourierdlem112  46139  fouriersw  46152  sge0tsms  46301  sge0resrnlem  46324  meadjiunlem  46386  omeunle  46437  isomenndlem  46451