Theorem fcomptf 29458

Description: Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6400. (Contributed by Thierry Arnoux, 30-Jun-2017.)

Hypothesis

Ref	Expression
fcomptf.1	|- F/_ x B

Assertion

Ref	Expression
fcomptf	|- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Distinct variable groups:   x, A   x, C   x, D   x, E

Allowed substitution hint:   B( x)

Proof of Theorem fcomptf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . . 5 |- F/_ x A
2		nfcv 2764	. . . . 5 |- F/_ x D
3		nfcv 2764	. . . . 5 |- F/_ x E
4	1, 2, 3	nff 6041	. . . 4 |- F/ x A : D --> E
5		fcomptf.1	. . . . 5 |- F/_ x B
6		nfcv 2764	. . . . 5 |- F/_ x C
7	5, 6, 2	nff 6041	. . . 4 |- F/ x B : C --> D
8	4, 7	nfan 1828	. . 3 |- F/ x ( A : D --> E /\ B : C --> D )
9		ffvelrn 6357	. . . . 5 |- ( ( B : C --> D /\ x e. C ) -> ( B ` x ) e. D )
10	9	adantll 750	. . . 4 |- ( ( ( A : D --> E /\ B : C --> D ) /\ x e. C ) -> ( B ` x ) e. D )
11	10	ex 450	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> ( x e. C -> ( B ` x ) e. D ) )
12	8, 11	ralrimi 2957	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> A. x e. C ( B ` x ) e. D )
13		ffn 6045	. . . 4 |- ( B : C --> D -> B Fn C )
14	13	adantl 482	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
15	5	dffn5f 6252	. . 3 |- ( B Fn C <-> B = ( x e. C |-> ( B ` x ) ) )
16	14, 15	sylib 208	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> B = ( x e. C |-> ( B ` x ) ) )
17		ffn 6045	. . . 4 |- ( A : D --> E -> A Fn D )
18	17	adantr 481	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
19		dffn5 6241	. . 3 |- ( A Fn D <-> A = ( y e. D |-> ( A ` y ) ) )
20	18, 19	sylib 208	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> A = ( y e. D |-> ( A ` y ) ) )
21		fveq2 6191	. 2 |- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
22	12, 16, 20, 21	fmptcof 6397	1 |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   |-> cmpt 4729   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  ofoprabco  29464