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Theorem ofcfval 28931
Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1  |-  ( ph  ->  F  Fn  A )
ofcfval.2  |-  ( ph  ->  A  e.  V )
ofcfval.3  |-  ( ph  ->  C  e.  W )
ofcfval.6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
Assertion
Ref Expression
ofcfval  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Distinct variable groups:    x, C    x, F    x, R    ph, x
Allowed substitution hints:    A( x)    B( x)    V( x)    W( x)

Proof of Theorem ofcfval
Dummy variables  f 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofc 28929 . . . 4  |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) R c ) ) )
21a1i 11 . . 3  |-  ( ph  ->𝑓/𝑐 R  =  ( f  e.  _V ,  c  e.  _V  |->  ( x  e.  dom  f  |->  ( ( f `
 x ) R c ) ) ) )
3 simprl 765 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
f  =  F )
43dmeqd 5040 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  ->  dom  f  =  dom  F )
53fveq1d 5872 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
( f `  x
)  =  ( F `
 x ) )
6 simprr 767 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
c  =  C )
75, 6oveq12d 6313 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
( ( f `  x ) R c )  =  ( ( F `  x ) R C ) )
84, 7mpteq12dv 4484 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
( x  e.  dom  f  |->  ( ( f `
 x ) R c ) )  =  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) ) )
9 ofcfval.1 . . . 4  |-  ( ph  ->  F  Fn  A )
10 ofcfval.2 . . . 4  |-  ( ph  ->  A  e.  V )
11 fnex 6137 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
129, 10, 11syl2anc 667 . . 3  |-  ( ph  ->  F  e.  _V )
13 ofcfval.3 . . . 4  |-  ( ph  ->  C  e.  W )
14 elex 3056 . . . 4  |-  ( C  e.  W  ->  C  e.  _V )
1513, 14syl 17 . . 3  |-  ( ph  ->  C  e.  _V )
16 fndm 5680 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
179, 16syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  A )
1817, 10eqeltrd 2531 . . . 4  |-  ( ph  ->  dom  F  e.  V
)
19 mptexg 6140 . . . 4  |-  ( dom 
F  e.  V  -> 
( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  e.  _V )
2018, 19syl 17 . . 3  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  e.  _V )
212, 8, 12, 15, 20ovmpt2d 6429 . 2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  dom  F  |->  ( ( F `  x ) R C ) ) )
2217eleq2d 2516 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
2322pm5.32i 643 . . . . 5  |-  ( (
ph  /\  x  e.  dom  F )  <->  ( ph  /\  x  e.  A ) )
24 ofcfval.6 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
2523, 24sylbi 199 . . . 4  |-  ( (
ph  /\  x  e.  dom  F )  ->  ( F `  x )  =  B )
2625oveq1d 6310 . . 3  |-  ( (
ph  /\  x  e.  dom  F )  ->  (
( F `  x
) R C )  =  ( B R C ) )
2717, 26mpteq12dva 4483 . 2  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  =  ( x  e.  A  |->  ( B R C ) ) )
2821, 27eqtrd 2487 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   _Vcvv 3047    |-> cmpt 4464   dom cdm 4837    Fn wfn 5580   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297  ∘𝑓/𝑐cofc 28928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-ofc 28929
This theorem is referenced by:  ofcval  28932  ofcfn  28933  ofcfeqd2  28934  ofcf  28936  ofcfval2  28937  ofcc  28939  ofcof  28940
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