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Theorem ofcof 28775
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 5746 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 17 . . 3  |-  ( ph  ->  F  Fn  A )
4 ofcof.2 . . 3  |-  ( ph  ->  A  e.  V )
5 ofcof.3 . . 3  |-  ( ph  ->  C  e.  W )
6 eqidd 2430 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
73, 4, 5, 6ofcfval 28766 . 2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
8 fnconstg 5788 . . . 4  |-  ( C  e.  W  ->  ( A  X.  { C }
)  Fn  A )
95, 8syl 17 . . 3  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
10 inidm 3677 . . 3  |-  ( A  i^i  A )  =  A
11 fvconst2g 6133 . . . 4  |-  ( ( C  e.  W  /\  x  e.  A )  ->  ( ( A  X.  { C } ) `  x )  =  C )
125, 11sylan 473 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  { C } ) `  x
)  =  C )
133, 9, 4, 4, 10, 6, 12offval 6552 . 2  |-  ( ph  ->  ( F  oF R ( A  X.  { C } ) )  =  ( x  e.  A  |->  ( ( F `
 x ) R C ) ) )
147, 13eqtr4d 2473 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F  oF R ( A  X.  { C } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {csn 4002    |-> cmpt 4484    X. cxp 4852    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543  ∘𝑓/𝑐cofc 28763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofc 28764
This theorem is referenced by:  ofcccat  29226
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