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

Theorem fcompt 7128
Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcompt ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸

Proof of Theorem fcompt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ffvelcdm 7081 . . 3 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
21adantll 713 . 2 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
3 ffn 6715 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
43adantl 483 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
5 dffn5 6948 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
64, 5sylib 217 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
7 ffn 6715 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
87adantr 482 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
9 dffn5 6948 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
108, 9sylib 217 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
11 fveq2 6889 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
122, 6, 10, 11fmptco 7124 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cmpt 5231  ccom 5680   Fn wfn 6536  wf 6537  cfv 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549
This theorem is referenced by:  2fvcoidd  7292  revco  14782  repsco  14788  caucvgrlem2  15618  fucidcl  17915  fucsect  17922  dfinito3  17952  dftermo3  17953  prf1st  18153  prf2nd  18154  curfcl  18182  yonedalem4c  18227  yonedalem3b  18229  yonedainv  18231  frmdup3  18745  smndex1gid  18781  efginvrel1  19591  frgpup3lem  19640  frgpup3  19641  dprdfinv  19884  grpvlinv  21889  grpvrinv  21890  mhmvlin  21891  chcoeffeqlem  22379  prdstps  23125  imasdsf1olem  23871  gamcvg2lem  26553  cofmpt2  31846  meascnbl  33206  elmrsubrn  34500  mzprename  41473  mendassa  41922  fcomptss  43888  mulc1cncfg  44292  expcnfg  44294  cncficcgt0  44591  fprodsubrecnncnvlem  44610  fprodaddrecnncnvlem  44612  dvsinax  44616  dirkercncflem2  44807  fourierdlem18  44828  fourierdlem53  44862  fourierdlem93  44902  fourierdlem101  44910  fourierdlem111  44920  sge0resrnlem  45106  omeiunle  45220  ovolval3  45350  amgmwlem  47803
  Copyright terms: Public domain W3C validator