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Theorem ofcfval 28357
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 28355 . . . 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 756 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
f  =  F )
43dmeqd 5215 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  ->  dom  f  =  dom  F )
53fveq1d 5874 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
( f `  x
)  =  ( F `
 x ) )
6 simprr 757 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
c  =  C )
75, 6oveq12d 6314 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  c  =  C ) )  -> 
( ( f `  x ) R c )  =  ( ( F `  x ) R C ) )
84, 7mpteq12dv 4535 . . 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 6140 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
129, 10, 11syl2anc 661 . . 3  |-  ( ph  ->  F  e.  _V )
13 ofcfval.3 . . . 4  |-  ( ph  ->  C  e.  W )
14 elex 3118 . . . 4  |-  ( C  e.  W  ->  C  e.  _V )
1513, 14syl 16 . . 3  |-  ( ph  ->  C  e.  _V )
16 fndm 5686 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
179, 16syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  A )
1817, 10eqeltrd 2545 . . . 4  |-  ( ph  ->  dom  F  e.  V
)
19 mptexg 6143 . . . 4  |-  ( dom 
F  e.  V  -> 
( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  e.  _V )
2018, 19syl 16 . . 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 2527 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
2322pm5.32i 637 . . . . 5  |-  ( (
ph  /\  x  e.  dom  F )  <->  ( ph  /\  x  e.  A ) )
24 ofcfval.6 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
2523, 24sylbi 195 . . . 4  |-  ( (
ph  /\  x  e.  dom  F )  ->  ( F `  x )  =  B )
2625oveq1d 6311 . . 3  |-  ( (
ph  /\  x  e.  dom  F )  ->  (
( F `  x
) R C )  =  ( B R C ) )
2717, 26mpteq12dva 4534 . 2  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( ( F `  x ) R C ) )  =  ( x  e.  A  |->  ( B R C ) ) )
2821, 27eqtrd 2498 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    |-> cmpt 4515   dom cdm 5008    Fn wfn 5589   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298  ∘𝑓/𝑐cofc 28354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-ofc 28355
This theorem is referenced by:  ofcval  28358  ofcfn  28359  ofcfeqd2  28360  ofcf  28362  ofcfval2  28363  ofcc  28365  ofcof  28366
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