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Theorem fcompt 7061
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 7015 . . 3 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
21adantll 711 . 2 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
3 ffn 6651 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
43adantl 482 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
5 dffn5 6884 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
64, 5sylib 217 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
7 ffn 6651 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
87adantr 481 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
9 dffn5 6884 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
108, 9sylib 217 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
11 fveq2 6825 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
122, 6, 10, 11fmptco 7057 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  cmpt 5175  ccom 5624   Fn wfn 6474  wf 6475  cfv 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487
This theorem is referenced by:  2fvcoidd  7225  revco  14646  repsco  14652  caucvgrlem2  15485  fucidcl  17780  fucsect  17787  dfinito3  17817  dftermo3  17818  prf1st  18018  prf2nd  18019  curfcl  18047  yonedalem4c  18092  yonedalem3b  18094  yonedainv  18096  frmdup3  18602  smndex1gid  18638  efginvrel1  19429  frgpup3lem  19478  frgpup3  19479  dprdfinv  19717  grpvlinv  21650  grpvrinv  21651  mhmvlin  21652  chcoeffeqlem  22140  prdstps  22886  imasdsf1olem  23632  gamcvg2lem  26314  cofmpt2  31256  meascnbl  32485  elmrsubrn  33781  mzprename  40841  mendassa  41290  fcomptss  43078  mulc1cncfg  43474  expcnfg  43476  cncficcgt0  43773  fprodsubrecnncnvlem  43792  fprodaddrecnncnvlem  43794  dvsinax  43798  dirkercncflem2  43989  fourierdlem18  44010  fourierdlem53  44044  fourierdlem93  44084  fourierdlem101  44092  fourierdlem111  44102  sge0resrnlem  44286  omeiunle  44400  ovolval3  44530  amgmwlem  46865
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