Theorem feqresmpt 6833

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 6638	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6832	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6788	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5182	. 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 1542   ⊆ wss 3892   ↦ cmpt 5162   ↾ cres 5591  ⟶wf 6427  ‘cfv 6431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-fv 6439

This theorem is referenced by:  pwfseqlem5  10418  pfxres  14388  gsumpt  19559  dpjidcl  19657  regsumsupp  20823  tsmsxplem2  23301  dvmulbr  25099  dvlip  25153  lhop1lem  25173  loglesqrt  25907  jensenlem1  26132  jensen  26134  amgm  26136  ushgredgedg  27592  ushgredgedgloop  27594  mgcf1o  31275  gsumzresunsn  31308  gsumpart  31309  gsumhashmul  31310  gsumle  31344  coinflippv  32444  fdvposlt  32573  fdvposle  32575  logdivsqrle  32624  ftc1cnnclem  35842  dvasin  35855  dvacos  35856  dvreasin  35857  dvreacos  35858  areacirclem1  35859  dvrelog2  40067  dvrelog3  40068  limsupvaluz2  43248  supcnvlimsup  43250  itgperiod  43491  fourierdlem69  43685  fourierdlem73  43689  fourierdlem74  43690  fourierdlem75  43691  fourierdlem76  43692  fourierdlem81  43697  fourierdlem85  43701  fourierdlem88  43704  fourierdlem92  43708  fourierdlem97  43713  fourierdlem100  43716  fourierdlem101  43717  fourierdlem103  43719  fourierdlem104  43720  fourierdlem107  43723  fourierdlem111  43727  fourierdlem112  43728  fouriersw  43741  sge0tsms  43887  sge0resrnlem  43910  meadjiunlem  43972  omeunle  44023  isomenndlem  44037