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Theorem ofcfval3 27926
Description: General value of  ( F𝑓/𝑐 R C ) with no assumptions on functionality of  F. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
ofcfval3  |-  ( ( F  e.  V  /\  C  e.  W )  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  dom  F  |->  ( ( F `  x ) R C ) ) )
Distinct variable groups:    x, C    x, F    x, R
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem ofcfval3
Dummy variables  f 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3127 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 465 . 2  |-  ( ( F  e.  V  /\  C  e.  W )  ->  F  e.  _V )
3 elex 3127 . . 3  |-  ( C  e.  W  ->  C  e.  _V )
43adantl 466 . 2  |-  ( ( F  e.  V  /\  C  e.  W )  ->  C  e.  _V )
5 dmexg 6726 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 mptexg 6141 . . . 4  |-  ( dom 
F  e.  _V  ->  ( x  e.  dom  F  |->  ( ( F `  x ) R C ) )  e.  _V )
75, 6syl 16 . . 3  |-  ( F  e.  V  ->  (
x  e.  dom  F  |->  ( ( F `  x ) R C ) )  e.  _V )
87adantr 465 . 2  |-  ( ( F  e.  V  /\  C  e.  W )  ->  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  e.  _V )
9 simpl 457 . . . . 5  |-  ( ( f  =  F  /\  c  =  C )  ->  f  =  F )
109dmeqd 5211 . . . 4  |-  ( ( f  =  F  /\  c  =  C )  ->  dom  f  =  dom  F )
119fveq1d 5874 . . . . 5  |-  ( ( f  =  F  /\  c  =  C )  ->  ( f `  x
)  =  ( F `
 x ) )
12 simpr 461 . . . . 5  |-  ( ( f  =  F  /\  c  =  C )  ->  c  =  C )
1311, 12oveq12d 6313 . . . 4  |-  ( ( f  =  F  /\  c  =  C )  ->  ( ( f `  x ) R c )  =  ( ( F `  x ) R C ) )
1410, 13mpteq12dv 4531 . . 3  |-  ( ( f  =  F  /\  c  =  C )  ->  ( x  e.  dom  f  |->  ( ( f `
 x ) R c ) )  =  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) ) )
15 df-ofc 27920 . . 3  |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) R c ) ) )
1614, 15ovmpt2ga 6427 . 2  |-  ( ( F  e.  _V  /\  C  e.  _V  /\  (
x  e.  dom  F  |->  ( ( F `  x ) R C ) )  e.  _V )  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  dom  F  |->  ( ( F `  x ) R C ) ) )
172, 4, 8, 16syl3anc 1228 1  |-  ( ( F  e.  V  /\  C  e.  W )  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  dom  F  |->  ( ( F `  x ) R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    |-> cmpt 4511   dom cdm 5005   ` cfv 5594  (class class class)co 6295  ∘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-8 1769  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  ax-un 6587
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-ofc 27920
This theorem is referenced by:  ofcfval4  27929  measdivcst  28021
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