Theorem feqresmpt 6954

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 6749	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6953	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6904	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5208	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2816	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3906   ↦ cmpt 5194   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548

This theorem is used by:  pwfseqlem5  10659  pfxres  14734  gsumpt  20055  dpjidcl  20153  gsumle  20238  regsumsupp  21801  tsmsxplem2  24340  dvmulbr  26127  dvlip  26181  lhop1lem  26201  loglesqrt  26955  jensenlem1  27180  jensen  27182  amgm  27184  ushgredgedg  29608  ushgredgedgloop  29610  fisuppov1  33057  fmptunsnop  33074  mgcf1o  33346  gsumfs2d  33404  gsumzresunsn  33405  gsumpart  33406  gsumhashmul  33410  rprmdvdsprod  33847  coinflippv  34898  fdvposlt  35010  fdvposle  35012  logdivsqrle  35061  ftc1cnnclem  38375  dvasin  38388  dvacos  38389  dvreasin  38390  dvreacos  38391  areacirclem1  38392  dvrelog2  42864  dvrelog3  42865  aks6d1c2  42930  aks6d1c6lem3  42972  readvrec2  43155  readvrec  43156  resuppsinopn  43157  cantnf2  44085  limsupvaluz2  46485  supcnvlimsup  46487  itgperiod  46728  fourierdlem69  46922  fourierdlem73  46926  fourierdlem74  46927  fourierdlem75  46928  fourierdlem76  46929  fourierdlem81  46934  fourierdlem85  46938  fourierdlem88  46941  fourierdlem92  46945  fourierdlem97  46950  fourierdlem100  46953  fourierdlem101  46954  fourierdlem103  46956  fourierdlem104  46957  fourierdlem107  46960  fourierdlem111  46964  fourierdlem112  46965  fouriersw  46978  sge0tsms  47127  sge0resrnlem  47150  meadjiunlem  47212  omeunle  47263  isomenndlem  47277