ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnofval Unicode version

Theorem fnofval 5752
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 2083 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2083 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 5750 . . . 4  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5211 . . 3  |-  ( ph  ->  ( ( F  oF R G ) `
 X )  =  ( ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) ) `  X ) )
109adantr 270 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
11 simpr 108 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  X  e.  S )
12 ofval.8 . . . . 5  |-  ( ph  ->  R  Fn  ( U  X.  V ) )
1312adantr 270 . . . 4  |-  ( (
ph  /\  X  e.  S )  ->  R  Fn  ( U  X.  V
) )
14 ofval.9 . . . . . 6  |-  ( ph  ->  C  e.  U )
1514adantr 270 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  C  e.  U )
16 inss1 3193 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
175, 16eqsstr3i 3031 . . . . . . . 8  |-  S  C_  A
1817sseli 2996 . . . . . . 7  |-  ( X  e.  S  ->  X  e.  A )
19 ofval.6 . . . . . . 7  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2018, 19sylan2 280 . . . . . 6  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  =  C )
2120eleq1d 2148 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
)  e.  U  <->  C  e.  U ) )
2215, 21mpbird 165 . . . 4  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  e.  U )
23 ofval.10 . . . . . 6  |-  ( ph  ->  D  e.  V )
2423adantr 270 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  D  e.  V )
25 inss2 3194 . . . . . . . . 9  |-  ( A  i^i  B )  C_  B
265, 25eqsstr3i 3031 . . . . . . . 8  |-  S  C_  B
2726sseli 2996 . . . . . . 7  |-  ( X  e.  S  ->  X  e.  B )
28 ofval.7 . . . . . . 7  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2927, 28sylan2 280 . . . . . 6  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  =  D )
3029eleq1d 2148 . . . . 5  |-  ( (
ph  /\  X  e.  S )  ->  (
( G `  X
)  e.  V  <->  D  e.  V ) )
3124, 30mpbird 165 . . . 4  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  e.  V )
32 fnovex 5569 . . . 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 1170 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  e.  _V )
34 fveq2 5209 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
35 fveq2 5209 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
3634, 35oveq12d 5561 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
37 eqid 2082 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
3836, 37fvmptg 5280 . . 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 403 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
4020, 29oveq12d 5561 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
4110, 39, 403eqtrd 2118 1  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( C R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602    i^i cin 2973    |-> cmpt 3847    X. cxp 4369    Fn wfn 4927   ` cfv 4932  (class class class)co 5543    oFcof 5741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-of 5743
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator