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Theorem offveq 6182
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 3454 . . 3  |-  ( A  i^i  A )  =  A
51, 2, 3, 3, 4offn 6173 . 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 6171 . . 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 2390 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( F  o F R G ) `  x )  =  ( H `  x ) )
125, 6, 11eqfnfvd 5705 1  |-  ( ph  ->  ( F  o F R G )  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    Fn wfn 5329   ` cfv 5334  (class class class)co 5942    o Fcof 6160
This theorem is referenced by:  caofid0l  6189  caofid0r  6190  caofid1  6191  caofid2  6192  ofnegsub  9831  bddibl  19292  dvaddf  19389  plydivlem3  19773  ofsubid  26864  ofmul12  26865  ofdivrec  26866  ofdivcan4  26867  ofdivdiv2  26868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162
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