Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  feqresmpt Structured version   Visualization version   GIF version

Theorem feqresmpt 6713
 Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
feqmptd.1 (𝜑𝐹:𝐴𝐵)
feqresmpt.2 (𝜑𝐶𝐴)
Assertion
Ref Expression
feqresmpt (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem feqresmpt
StepHypRef Expression
1 feqmptd.1 . . . 4 (𝜑𝐹:𝐴𝐵)
2 feqresmpt.2 . . . 4 (𝜑𝐶𝐴)
31, 2fssresd 6523 . . 3 (𝜑 → (𝐹𝐶):𝐶𝐵)
43feqmptd 6712 . 2 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)))
5 fvres 6668 . . 3 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
65mpteq2ia 5124 . 2 (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)) = (𝑥𝐶 ↦ (𝐹𝑥))
74, 6eqtrdi 2852 1 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ⊆ wss 3884   ↦ cmpt 5113   ↾ cres 5525  ⟶wf 6324  ‘cfv 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336 This theorem is referenced by:  pwfseqlem5  10078  pfxres  14036  gsumpt  19078  dpjidcl  19176  regsumsupp  20314  tsmsxplem2  22762  dvmulbr  24545  dvlip  24599  lhop1lem  24619  loglesqrt  25350  jensenlem1  25575  jensen  25577  amgm  25579  ushgredgedg  27022  ushgredgedgloop  27024  gsumzresunsn  30742  gsumpart  30743  gsumhashmul  30744  gsumle  30778  coinflippv  31849  fdvposlt  31978  fdvposle  31980  logdivsqrle  32029  ftc1cnnclem  35121  dvasin  35134  dvacos  35135  dvreasin  35136  dvreacos  35137  areacirclem1  35138  limsupvaluz2  42367  supcnvlimsup  42369  itgperiod  42610  fourierdlem69  42804  fourierdlem73  42808  fourierdlem74  42809  fourierdlem75  42810  fourierdlem76  42811  fourierdlem81  42816  fourierdlem85  42820  fourierdlem88  42823  fourierdlem92  42827  fourierdlem97  42832  fourierdlem100  42835  fourierdlem101  42836  fourierdlem103  42838  fourierdlem104  42839  fourierdlem107  42842  fourierdlem111  42846  fourierdlem112  42847  fouriersw  42860  sge0tsms  43006  sge0resrnlem  43029  meadjiunlem  43091  omeunle  43142  isomenndlem  43156
 Copyright terms: Public domain W3C validator