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Theorem ofcof 27931
Description: Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)
Hypotheses
Ref Expression
ofcof.1  |-  ( ph  ->  F : A --> B )
ofcof.2  |-  ( ph  ->  A  e.  V )
ofcof.3  |-  ( ph  ->  C  e.  W )
Assertion
Ref Expression
ofcof  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F  oF R ( A  X.  { C } ) ) )

Proof of Theorem ofcof
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofcof.1 . . . 4  |-  ( ph  ->  F : A --> B )
2 ffn 5737 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
4 ofcof.2 . . 3  |-  ( ph  ->  A  e.  V )
5 ofcof.3 . . 3  |-  ( ph  ->  C  e.  W )
6 eqidd 2468 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
73, 4, 5, 6ofcfval 27922 . 2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
8 fnconstg 5779 . . . 4  |-  ( C  e.  W  ->  ( A  X.  { C }
)  Fn  A )
95, 8syl 16 . . 3  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
10 inidm 3712 . . 3  |-  ( A  i^i  A )  =  A
11 fvconst2g 6125 . . . 4  |-  ( ( C  e.  W  /\  x  e.  A )  ->  ( ( A  X.  { C } ) `  x )  =  C )
125, 11sylan 471 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  { C } ) `  x
)  =  C )
133, 9, 4, 4, 10, 6, 12offval 6542 . 2  |-  ( ph  ->  ( F  oF R ( A  X.  { C } ) )  =  ( x  e.  A  |->  ( ( F `
 x ) R C ) ) )
147, 13eqtr4d 2511 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F  oF R ( A  X.  { C } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033    |-> cmpt 4511    X. cxp 5003    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533  ∘𝑓/𝑐cofc 27919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofc 27920
This theorem is referenced by:  ofcccat  28323
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