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Theorem feqresmpt 6289
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 6109 . . 3 (𝜑 → (𝐹𝐶):𝐶𝐵)
43feqmptd 6288 . 2 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)))
5 fvres 6245 . . 3 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
65mpteq2ia 4773 . 2 (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)) = (𝑥𝐶 ↦ (𝐹𝑥))
74, 6syl6eq 2701 1 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wss 3607  cmpt 4762  cres 5145  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  pwfseqlem5  9523  swrd0val  13466  gsumpt  18407  dpjidcl  18503  regsumsupp  20016  tsmsxplem2  22004  dvmulbr  23747  dvlip  23801  lhop1lem  23821  loglesqrt  24544  jensenlem1  24758  jensen  24760  amgm  24762  ushgredgedg  26166  ushgredgedgloop  26168  gsumle  29907  coinflippv  30673  fdvposlt  30805  fdvposle  30807  logdivsqrle  30856  ftc1cnnclem  33613  dvasin  33626  dvacos  33627  dvreasin  33628  dvreacos  33629  areacirclem1  33630  limsupvaluz2  40288  supcnvlimsup  40290  itgperiod  40515  fourierdlem69  40710  fourierdlem73  40714  fourierdlem74  40715  fourierdlem75  40716  fourierdlem76  40717  fourierdlem81  40722  fourierdlem85  40726  fourierdlem88  40729  fourierdlem92  40733  fourierdlem97  40738  fourierdlem100  40741  fourierdlem101  40742  fourierdlem103  40744  fourierdlem104  40745  fourierdlem107  40748  fourierdlem111  40752  fourierdlem112  40753  fouriersw  40766  sge0tsms  40915  sge0resrnlem  40938  meadjiunlem  41000  omeunle  41051  isomenndlem  41065  pfxres  41713
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