Theorem fcompt 7108

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 7056	. . 3 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
2	1	adantll 714	. 2 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
3		ffn 6691	. . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶)
4	3	adantl 481	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶)
5		dffn5 6922	. . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
6	4, 5	sylib 218	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
7		ffn 6691	. . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷)
8	7	adantr 480	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷)
9		dffn5 6922	. . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
10	8, 9	sylib 218	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
11		fveq2 6861	. 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥)))
12	2, 6, 10, 11	fmptco 7104	1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5191   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522

This theorem is referenced by:  2fvcoidd  7275  revco  14807  repsco  14813  caucvgrlem2  15648  fucidcl  17937  fucsect  17944  dfinito3  17974  dftermo3  17975  prf1st  18172  prf2nd  18173  curfcl  18200  yonedalem4c  18245  yonedalem3b  18247  yonedainv  18249  mhmvlin  18735  frmdup3  18801  smndex1gid  18837  efginvrel1  19665  frgpup3lem  19714  frgpup3  19715  dprdfinv  19958  grpvlinv  22292  grpvrinv  22293  chcoeffeqlem  22779  prdstps  23523  imasdsf1olem  24268  gamcvg2lem  26976  cofmpt2  32565  meascnbl  34216  elmrsubrn  35514  mzprename  42744  mendassa  43186  fcomptss  45204  mulc1cncfg  45594  expcnfg  45596  cncficcgt0  45893  fprodsubrecnncnvlem  45912  fprodaddrecnncnvlem  45914  dvsinax  45918  dirkercncflem2  46109  fourierdlem18  46130  fourierdlem53  46164  fourierdlem93  46204  fourierdlem101  46212  fourierdlem111  46222  sge0resrnlem  46408  omeiunle  46522  ovolval3  46652  fucorid2  49356  precofval2  49362  amgmwlem  49795