MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprab2co Structured version   Unicode version

Theorem oprab2co 6802
Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
Hypotheses
Ref Expression
oprab2co.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )
oprab2co.2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )
oprab2co.3  |-  F  =  ( x  e.  A ,  y  e.  B  |-> 
<. C ,  D >. )
oprab2co.4  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )
Assertion
Ref Expression
oprab2co  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Distinct variable groups:    x, y, A    x, B, y    x, M, y    x, R, y   
x, S, y
Allowed substitution hints:    C( x, y)    D( x, y)    F( x, y)    G( x, y)

Proof of Theorem oprab2co
StepHypRef Expression
1 oprab2co.1 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )
2 oprab2co.2 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )
3 opelxpi 4958 . . 3  |-  ( ( C  e.  R  /\  D  e.  S )  -> 
<. C ,  D >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 659 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. C ,  D >.  e.  ( R  X.  S
) )
5 oprab2co.3 . 2  |-  F  =  ( x  e.  A ,  y  e.  B  |-> 
<. C ,  D >. )
6 oprab2co.4 . . 3  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )
7 df-ov 6217 . . . . 5  |-  ( C M D )  =  ( M `  <. C ,  D >. )
87a1i 11 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( C M D )  =  ( M `
 <. C ,  D >. ) )
98mpt2eq3ia 6279 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )  =  ( x  e.  A ,  y  e.  B  |->  ( M `  <. C ,  D >. ) )
106, 9eqtri 2421 . 2  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( M `  <. C ,  D >. )
)
114, 5, 10oprabco 6801 1  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   <.cop 3963    X. cxp 4924    o. ccom 4930    Fn wfn 5504   ` cfv 5509  (class class class)co 6214    |-> cmpt2 6216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-1st 6717  df-2nd 6718
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator