Theorem fcompt 5276

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 o. B) = (x e. 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 5243	. . 3 |- ((B : C --> D /\ x e. C) -> (B ` x) e. D)
2	1	adantll 445	. 2 |- (((A : D --> E /\ B : C --> D) /\ x e. C) -> (B ` x) e. D)
3		ffn 4989	. . . 4 |- (B : C --> D -> B Fn C)
4	3	adantl 262	. . 3 |- ((A : D --> E /\ B : C --> D) -> B Fn C)
5		dffn5im 5162	. . 3 |- (B Fn C -> B = (x e. C |-> (B ` x)))
6	4, 5	syl 14	. 2 |- ((A : D --> E /\ B : C --> D) -> B = (x e. C |-> (B ` x)))
7		ffn 4989	. . . 4 |- (A : D --> E -> A Fn D)
8	7	adantr 261	. . 3 |- ((A : D --> E /\ B : C --> D) -> A Fn D)
9		dffn5im 5162	. . 3 |- (A Fn D -> A = (y e. D |-> (A ` y)))
10	8, 9	syl 14	. 2 |- ((A : D --> E /\ B : C --> D) -> A = (y e. D |-> (A ` y)))
11		fveq2 5121	. 2 |- (y = (B ` x) -> (A ` y) = (A ` (B ` x)))
12	2, 6, 10, 11	fmptco 5273	1 |- ((A : D --> E /\ B : C --> D) -> (A o. B) = (x e. C |-> (A ` (B ` x))))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390   |-> cmpt 3809   o. ccom 4292   Fn wfn 4840   -->wf 4841   ` cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853

This theorem is referenced by: (None)