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

Theorem offveq 6284
Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
offveq.1  |-  ( ph  ->  A  e.  V )
offveq.2  |-  ( ph  ->  F  Fn  A )
offveq.3  |-  ( ph  ->  G  Fn  A )
offveq.4  |-  ( ph  ->  H  Fn  A )
offveq.5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
offveq.6  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  C )
offveq.7  |-  ( (
ph  /\  x  e.  A )  ->  ( B R C )  =  ( H `  x
) )
Assertion
Ref Expression
offveq  |-  ( ph  ->  ( F  o F R G )  =  H )
Distinct variable groups:    x, A    x, F    x, G    x, H    ph, x    x, R
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem offveq
StepHypRef Expression
1 offveq.2 . . 3  |-  ( ph  ->  F  Fn  A )
2 offveq.3 . . 3  |-  ( ph  ->  G  Fn  A )
3 offveq.1 . . 3  |-  ( ph  ->  A  e.  V )
4 inidm 3510 . . 3  |-  ( A  i^i  A )  =  A
51, 2, 3, 3, 4offn 6275 . 2  |-  ( ph  ->  ( F  o F R G )  Fn  A )
6 offveq.4 . 2  |-  ( ph  ->  H  Fn  A )
7 offveq.5 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
8 offveq.6 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  C )
91, 2, 3, 3, 4, 7, 8ofval 6273 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F  o F R G ) `  x )  =  ( B R C ) )
10 offveq.7 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( B R C )  =  ( H `  x
) )
119, 10eqtrd 2436 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( F  o F R G ) `  x )  =  ( H `  x ) )
125, 6, 11eqfnfvd 5789 1  |-  ( ph  ->  ( F  o F R G )  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    Fn wfn 5408   ` cfv 5413  (class class class)co 6040    o Fcof 6262
This theorem is referenced by:  caofid0l  6291  caofid0r  6292  caofid1  6293  caofid2  6294  ofnegsub  9954  bddibl  19684  dvaddf  19781  plydivlem3  20165  ofsubid  27409  ofmul12  27410  ofdivrec  27411  ofdivcan4  27412  ofdivdiv2  27413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
  Copyright terms: Public domain W3C validator