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Theorem ofcfval3 26480
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 2979 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 462 . 2  |-  ( ( F  e.  V  /\  C  e.  W )  ->  F  e.  _V )
3 elex 2979 . . 3  |-  ( C  e.  W  ->  C  e.  _V )
43adantl 463 . 2  |-  ( ( F  e.  V  /\  C  e.  W )  ->  C  e.  _V )
5 dmexg 6508 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 mptexg 5944 . . . 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 462 . 2  |-  ( ( F  e.  V  /\  C  e.  W )  ->  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  e.  _V )
9 simpl 454 . . . . 5  |-  ( ( f  =  F  /\  c  =  C )  ->  f  =  F )
109dmeqd 5038 . . . 4  |-  ( ( f  =  F  /\  c  =  C )  ->  dom  f  =  dom  F )
119fveq1d 5690 . . . . 5  |-  ( ( f  =  F  /\  c  =  C )  ->  ( f `  x
)  =  ( F `
 x ) )
12 simpr 458 . . . . 5  |-  ( ( f  =  F  /\  c  =  C )  ->  c  =  C )
1311, 12oveq12d 6108 . . . 4  |-  ( ( f  =  F  /\  c  =  C )  ->  ( ( f `  x ) R c )  =  ( ( F `  x ) R C ) )
1410, 13mpteq12dv 4367 . . 3  |-  ( ( f  =  F  /\  c  =  C )  ->  ( x  e.  dom  f  |->  ( ( f `
 x ) R c ) )  =  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) ) )
15 df-ofc 26474 . . 3  |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) R c ) ) )
1614, 15ovmpt2ga 6219 . 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 1213 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 1364    e. wcel 1761   _Vcvv 2970    e. cmpt 4347   dom cdm 4836   ` cfv 5415  (class class class)co 6090  ∘𝑓/𝑐cofc 26473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-ofc 26474
This theorem is referenced by:  ofcfval4  26483  measdivcst  26575
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