Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fcomptf Structured version   Unicode version

Theorem fcomptf 28100
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 6074. (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.
StepHypRef Expression
1 nfcv 2591 . . . . 5  |-  F/_ x A
2 nfcv 2591 . . . . 5  |-  F/_ x D
3 nfcv 2591 . . . . 5  |-  F/_ x E
41, 2, 3nff 5742 . . . 4  |-  F/ x  A : D --> E
5 fcomptf.1 . . . . 5  |-  F/_ x B
6 nfcv 2591 . . . . 5  |-  F/_ x C
75, 6, 2nff 5742 . . . 4  |-  F/ x  B : C --> D
84, 7nfan 1986 . . 3  |-  F/ x
( A : D --> E  /\  B : C --> D )
9 ffvelrn 6035 . . . . 5  |-  ( ( B : C --> D  /\  x  e.  C )  ->  ( B `  x
)  e.  D )
109adantll 718 . . . 4  |-  ( ( ( A : D --> E  /\  B : C --> D )  /\  x  e.  C )  ->  ( B `  x )  e.  D )
1110ex 435 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( x  e.  C  ->  ( B `  x )  e.  D
) )
128, 11ralrimi 2832 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A. x  e.  C  ( B `  x )  e.  D )
13 ffn 5746 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
1413adantl 467 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
155dffn5f 5936 . . 3  |-  ( B  Fn  C  <->  B  =  ( x  e.  C  |->  ( B `  x
) ) )
1614, 15sylib 199 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
17 ffn 5746 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
1817adantr 466 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
19 dffn5 5926 . . 3  |-  ( A  Fn  D  <->  A  =  ( y  e.  D  |->  ( A `  y
) ) )
2018, 19sylib 199 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
21 fveq2 5881 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
2212, 16, 20, 21fmptcof 6072 1  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
 x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   F/_wnfc 2577    |-> cmpt 4484    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609
This theorem is referenced by:  ofoprabco  28107
  Copyright terms: Public domain W3C validator