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Theorem offveq 6546
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  oF 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 3692 . . 3  |-  ( A  i^i  A )  =  A
51, 2, 3, 3, 4offn 6536 . 2  |-  ( ph  ->  ( F  oF 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 6534 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F  oF R G ) `  x )  =  ( B R C ) )
10 offveq.7 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( B R C )  =  ( H `  x
) )
119, 10eqtrd 2484 . 2  |-  ( (
ph  /\  x  e.  A )  ->  (
( F  oF R G ) `  x )  =  ( H `  x ) )
125, 6, 11eqfnfvd 5969 1  |-  ( ph  ->  ( F  oF R G )  =  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    Fn wfn 5573   ` cfv 5578  (class class class)co 6281    oFcof 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525
This theorem is referenced by:  caofid0l  6553  caofid0r  6554  caofid1  6555  caofid2  6556  ofnegsub  10540  bddibl  22119  dvaddf  22218  plydivlem3  22563  ofsubid  31205  ofmul12  31206  ofdivrec  31207  ofdivcan4  31208  ofdivdiv2  31209
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