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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 7081 | . . 3 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) | |
2 | 1 | adantll 713 | . 2 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) |
3 | ffn 6715 | . . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶) | |
4 | 3 | adantl 483 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶) |
5 | dffn5 6948 | . . 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 6715 | . . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷) | |
8 | 7 | adantr 482 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷) |
9 | dffn5 6948 | . . 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 6889 | . 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥))) | |
12 | 2, 6, 10, 11 | fmptco 7124 | 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 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5231 ∘ ccom 5680 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 |
This theorem is referenced by: 2fvcoidd 7292 revco 14782 repsco 14788 caucvgrlem2 15618 fucidcl 17915 fucsect 17922 dfinito3 17952 dftermo3 17953 prf1st 18153 prf2nd 18154 curfcl 18182 yonedalem4c 18227 yonedalem3b 18229 yonedainv 18231 frmdup3 18745 smndex1gid 18781 efginvrel1 19591 frgpup3lem 19640 frgpup3 19641 dprdfinv 19884 grpvlinv 21889 grpvrinv 21890 mhmvlin 21891 chcoeffeqlem 22379 prdstps 23125 imasdsf1olem 23871 gamcvg2lem 26553 cofmpt2 31846 meascnbl 33206 elmrsubrn 34500 mzprename 41473 mendassa 41922 fcomptss 43888 mulc1cncfg 44292 expcnfg 44294 cncficcgt0 44591 fprodsubrecnncnvlem 44610 fprodaddrecnncnvlem 44612 dvsinax 44616 dirkercncflem2 44807 fourierdlem18 44828 fourierdlem53 44862 fourierdlem93 44902 fourierdlem101 44910 fourierdlem111 44920 sge0resrnlem 45106 omeiunle 45220 ovolval3 45350 amgmwlem 47803 |
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