Theorem fcompt 7130

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:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

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.

Step	Hyp	Ref	Expression
1		ffvelcdm 7083	. . 3 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
2	1	adantll 712	. 2 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
3		ffn 6717	. . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶)
4	3	adantl 482	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶)
5		dffn5 6950	. . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
6	4, 5	sylib 217	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
7		ffn 6717	. . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷)
8	7	adantr 481	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷)
9		dffn5 6950	. . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
10	8, 9	sylib 217	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
11		fveq2 6891	. 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥)))
12	2, 6, 10, 11	fmptco 7126	1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5231   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is referenced by:  2fvcoidd  7294  revco  14784  repsco  14790  caucvgrlem2  15620  fucidcl  17917  fucsect  17924  dfinito3  17954  dftermo3  17955  prf1st  18155  prf2nd  18156  curfcl  18184  yonedalem4c  18229  yonedalem3b  18231  yonedainv  18233  frmdup3  18747  smndex1gid  18783  efginvrel1  19595  frgpup3lem  19644  frgpup3  19645  dprdfinv  19888  grpvlinv  21896  grpvrinv  21897  mhmvlin  21898  chcoeffeqlem  22386  prdstps  23132  imasdsf1olem  23878  gamcvg2lem  26560  cofmpt2  31853  meascnbl  33212  elmrsubrn  34506  mzprename  41477  mendassa  41926  fcomptss  43892  mulc1cncfg  44295  expcnfg  44297  cncficcgt0  44594  fprodsubrecnncnvlem  44613  fprodaddrecnncnvlem  44615  dvsinax  44619  dirkercncflem2  44810  fourierdlem18  44831  fourierdlem53  44865  fourierdlem93  44905  fourierdlem101  44913  fourierdlem111  44923  sge0resrnlem  45109  omeiunle  45223  ovolval3  45353  amgmwlem  47839