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Theorem fnofval 5923
Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1  |-  ( ph  ->  F  Fn  A )
offval.2  |-  ( ph  ->  G  Fn  B )
offval.3  |-  ( ph  ->  A  e.  V )
offval.4  |-  ( ph  ->  B  e.  W )
offval.5  |-  ( A  i^i  B )  =  S
ofval.6  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
ofval.7  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
ofval.8  |-  ( ph  ->  R  Fn  ( U  X.  V ) )
ofval.9  |-  ( ph  ->  C  e.  U )
ofval.10  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
fnofval  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( C R D ) )

Proof of Theorem fnofval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . 5  |-  ( ph  ->  F  Fn  A )
2 offval.2 . . . . 5  |-  ( ph  ->  G  Fn  B )
3 offval.3 . . . . 5  |-  ( ph  ->  A  e.  V )
4 offval.4 . . . . 5  |-  ( ph  ->  B  e.  W )
5 offval.5 . . . . 5  |-  ( A  i^i  B )  =  S
6 eqidd 2101 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2101 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 5921 . . . 4  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5355 . . 3  |-  ( ph  ->  ( ( F  oF R G ) `
 X )  =  ( ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) ) `  X ) )
109adantr 272 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
11 simpr 109 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  X  e.  S )
12 ofval.8 . . . . 5  |-  ( ph  ->  R  Fn  ( U  X.  V ) )
1312adantr 272 . . . 4  |-  ( (
ph  /\  X  e.  S )  ->  R  Fn  ( U  X.  V
) )
14 ofval.9 . . . . . 6  |-  ( ph  ->  C  e.  U )
1514adantr 272 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  C  e.  U )
16 inss1 3243 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
175, 16eqsstr3i 3080 . . . . . . . 8  |-  S  C_  A
1817sseli 3043 . . . . . . 7  |-  ( X  e.  S  ->  X  e.  A )
19 ofval.6 . . . . . . 7  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2018, 19sylan2 282 . . . . . 6  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  =  C )
2120eleq1d 2168 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
)  e.  U  <->  C  e.  U ) )
2215, 21mpbird 166 . . . 4  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  e.  U )
23 ofval.10 . . . . . 6  |-  ( ph  ->  D  e.  V )
2423adantr 272 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  D  e.  V )
25 inss2 3244 . . . . . . . . 9  |-  ( A  i^i  B )  C_  B
265, 25eqsstr3i 3080 . . . . . . . 8  |-  S  C_  B
2726sseli 3043 . . . . . . 7  |-  ( X  e.  S  ->  X  e.  B )
28 ofval.7 . . . . . . 7  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2927, 28sylan2 282 . . . . . 6  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  =  D )
3029eleq1d 2168 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  (
( G `  X
)  e.  V  <->  D  e.  V ) )
3124, 30mpbird 166 . . . 4  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  e.  V )
32 fnovex 5736 . . . 4  |-  ( ( R  Fn  ( U  X.  V )  /\  ( F `  X )  e.  U  /\  ( G `  X )  e.  V )  ->  (
( F `  X
) R ( G `
 X ) )  e.  _V )
3313, 22, 31, 32syl3anc 1184 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  e.  _V )
34 fveq2 5353 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
35 fveq2 5353 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
3634, 35oveq12d 5724 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
37 eqid 2100 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
3836, 37fvmptg 5429 . . 3  |-  ( ( X  e.  S  /\  ( ( F `  X ) R ( G `  X ) )  e.  _V )  ->  ( ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
3911, 33, 38syl2anc 406 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
4020, 29oveq12d 5724 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
4110, 39, 403eqtrd 2136 1  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( C R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448   _Vcvv 2641    i^i cin 3020    |-> cmpt 3929    X. cxp 4475    Fn wfn 5054   ` cfv 5059  (class class class)co 5706    oFcof 5912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-of 5914
This theorem is referenced by: (None)
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