| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fcompt | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| fcompt | ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ffvelcdm 7101 | . . 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 6736 | . . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶) | |
| 4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶) |
| 5 | dffn5 6967 | . . 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 6736 | . . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷) | |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷) |
| 9 | dffn5 6967 | . . 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 6906 | . 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥))) | |
| 12 | 2, 6, 10, 11 | fmptco 7149 | 1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: 2fvcoidd 7317 revco 14873 repsco 14879 caucvgrlem2 15711 fucidcl 18013 fucsect 18020 dfinito3 18050 dftermo3 18051 prf1st 18249 prf2nd 18250 curfcl 18277 yonedalem4c 18322 yonedalem3b 18324 yonedainv 18326 mhmvlin 18814 frmdup3 18880 smndex1gid 18916 efginvrel1 19746 frgpup3lem 19795 frgpup3 19796 dprdfinv 20039 grpvlinv 22402 grpvrinv 22403 chcoeffeqlem 22891 prdstps 23637 imasdsf1olem 24383 gamcvg2lem 27102 cofmpt2 32644 meascnbl 34220 elmrsubrn 35525 mzprename 42760 mendassa 43202 fcomptss 45208 mulc1cncfg 45604 expcnfg 45606 cncficcgt0 45903 fprodsubrecnncnvlem 45922 fprodaddrecnncnvlem 45924 dvsinax 45928 dirkercncflem2 46119 fourierdlem18 46140 fourierdlem53 46174 fourierdlem93 46214 fourierdlem101 46222 fourierdlem111 46232 sge0resrnlem 46418 omeiunle 46532 ovolval3 46662 fucorid2 49058 precofval2 49064 amgmwlem 49321 |
| Copyright terms: Public domain | W3C validator |