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Theorem feqresmpt 6145
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 5969 . . 3 (𝜑 → (𝐹𝐶):𝐶𝐵)
43feqmptd 6144 . 2 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)))
5 fvres 6102 . . 3 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
65mpteq2ia 4662 . 2 (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)) = (𝑥𝐶 ↦ (𝐹𝑥))
74, 6syl6eq 2659 1 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wss 3539  cmpt 4637  cres 5030  wf 5786  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798
This theorem is referenced by:  pwfseqlem5  9341  swrd0val  13219  gsumpt  18130  dpjidcl  18226  regsumsupp  19732  tsmsxplem2  21709  dvmulbr  23425  dvlip  23477  lhop1lem  23497  loglesqrt  24216  jensenlem1  24430  jensen  24432  amgm  24434  gsumle  28916  coinflippv  29678  ftc1cnnclem  32449  dvasin  32462  dvacos  32463  dvreasin  32464  dvreacos  32465  areacirclem1  32466  itgperiod  38670  fourierdlem69  38865  fourierdlem73  38869  fourierdlem74  38870  fourierdlem75  38871  fourierdlem76  38872  fourierdlem81  38877  fourierdlem85  38881  fourierdlem88  38884  fourierdlem92  38888  fourierdlem97  38893  fourierdlem100  38896  fourierdlem101  38897  fourierdlem103  38899  fourierdlem104  38900  fourierdlem107  38903  fourierdlem111  38907  fourierdlem112  38908  fouriersw  38921  sge0tsms  39070  sge0resrnlem  39093  meadjiunlem  39155  omeunle  39203  isomenndlem  39217  pfxres  40049  ushgredgedga  40451  ushgredgedgaloop  40453
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