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Theorem fcompt 7152
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 7100 . . 3 ((𝐵:𝐶𝐷𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
21adantll 714 . 2 (((𝐴:𝐷𝐸𝐵:𝐶𝐷) ∧ 𝑥𝐶) → (𝐵𝑥) ∈ 𝐷)
3 ffn 6736 . . . 4 (𝐵:𝐶𝐷𝐵 Fn 𝐶)
43adantl 481 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 Fn 𝐶)
5 dffn5 6966 . . 3 (𝐵 Fn 𝐶𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
64, 5sylib 218 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐵 = (𝑥𝐶 ↦ (𝐵𝑥)))
7 ffn 6736 . . . 4 (𝐴:𝐷𝐸𝐴 Fn 𝐷)
87adantr 480 . . 3 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 Fn 𝐷)
9 dffn5 6966 . . 3 (𝐴 Fn 𝐷𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
108, 9sylib 218 . 2 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → 𝐴 = (𝑦𝐷 ↦ (𝐴𝑦)))
11 fveq2 6906 . 2 (𝑦 = (𝐵𝑥) → (𝐴𝑦) = (𝐴‘(𝐵𝑥)))
122, 6, 10, 11fmptco 7148 1 ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cmpt 5230  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570
This theorem is referenced by:  2fvcoidd  7316  revco  14869  repsco  14875  caucvgrlem2  15707  fucidcl  18021  fucsect  18028  dfinito3  18058  dftermo3  18059  prf1st  18259  prf2nd  18260  curfcl  18288  yonedalem4c  18333  yonedalem3b  18335  yonedainv  18337  mhmvlin  18826  frmdup3  18892  smndex1gid  18928  efginvrel1  19760  frgpup3lem  19809  frgpup3  19810  dprdfinv  20053  grpvlinv  22417  grpvrinv  22418  chcoeffeqlem  22906  prdstps  23652  imasdsf1olem  24398  gamcvg2lem  27116  cofmpt2  32650  meascnbl  34199  elmrsubrn  35504  mzprename  42736  mendassa  43178  fcomptss  45145  mulc1cncfg  45544  expcnfg  45546  cncficcgt0  45843  fprodsubrecnncnvlem  45862  fprodaddrecnncnvlem  45864  dvsinax  45868  dirkercncflem2  46059  fourierdlem18  46080  fourierdlem53  46114  fourierdlem93  46154  fourierdlem101  46162  fourierdlem111  46172  sge0resrnlem  46358  omeiunle  46472  ovolval3  46602  amgmwlem  49032
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