Theorem fcompt 6758

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		ffvelrn 6714	. . 3 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
2	1	adantll 710	. 2 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷)
3		ffn 6382	. . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶)
4	3	adantl 482	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶)
5		dffn5 6592	. . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
6	4, 5	sylib 219	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥)))
7		ffn 6382	. . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷)
8	7	adantr 481	. . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷)
9		dffn5 6592	. . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
10	8, 9	sylib 219	. 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦)))
11		fveq2 6538	. 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥)))
12	2, 6, 10, 11	fmptco 6754	1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ↦ cmpt 5041   ∘ ccom 5447   Fn wfn 6220  ⟶wf 6221  ‘cfv 6225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233

This theorem is referenced by:  2fvcoidd  6918  revco  14032  repsco  14038  caucvgrlem2  14865  fucidcl  17064  fucsect  17071  prf1st  17283  prf2nd  17284  curfcl  17311  yonedalem4c  17356  yonedalem3b  17358  yonedainv  17360  frmdup3  17843  efginvrel1  18581  frgpup3lem  18630  frgpup3  18631  dprdfinv  18858  grpvlinv  20688  grpvrinv  20689  mhmvlin  20690  chcoeffeqlem  21177  prdstps  21921  imasdsf1olem  22666  gamcvg2lem  25318  cofmpt2  30069  meascnbl  31095  elmrsubrn  32376  mzprename  38850  mendassa  39298  fcomptss  41025  mulc1cncfg  41431  expcnfg  41433  cncficcgt0  41732  fprodsubrecnncnvlem  41752  fprodaddrecnncnvlem  41754  dvsinax  41758  dirkercncflem2  41951  fourierdlem18  41972  fourierdlem53  42006  fourierdlem93  42046  fourierdlem101  42054  fourierdlem111  42064  sge0resrnlem  42247  omeiunle  42361  ovolval3  42491  amgmwlem  44403