Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofmpteq Unicode version

Theorem ofmpteq 26165
Description: Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Assertion
Ref Expression
ofmpteq  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( (
x  e.  A  |->  B )  o F R ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( B R C ) ) )
Distinct variable groups:    x, A    x, R
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem ofmpteq
StepHypRef Expression
1 simp1 960 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  A  e.  V )
2 simpr 449 . . . 4  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  a  e.  A )
3 simpl2 964 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  (
x  e.  A  |->  B )  Fn  A )
4 eqid 2258 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54mptfng 5307 . . . . 5  |-  ( A. x  e.  A  B  e.  _V  <->  ( x  e.  A  |->  B )  Fn  A )
63, 5sylibr 205 . . . 4  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  A. x  e.  A  B  e.  _V )
7 nfcsb1v 3088 . . . . . 6  |-  F/_ x [_ a  /  x ]_ B
87nfel1 2404 . . . . 5  |-  F/ x [_ a  /  x ]_ B  e.  _V
9 csbeq1a 3064 . . . . . 6  |-  ( x  =  a  ->  B  =  [_ a  /  x ]_ B )
109eleq1d 2324 . . . . 5  |-  ( x  =  a  ->  ( B  e.  _V  <->  [_ a  /  x ]_ B  e.  _V ) )
118, 10rcla4 2853 . . . 4  |-  ( a  e.  A  ->  ( A. x  e.  A  B  e.  _V  ->  [_ a  /  x ]_ B  e.  _V )
)
122, 6, 11sylc 58 . . 3  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  [_ a  /  x ]_ B  e. 
_V )
13 simpl3 965 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  (
x  e.  A  |->  C )  Fn  A )
14 eqid 2258 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1514mptfng 5307 . . . . 5  |-  ( A. x  e.  A  C  e.  _V  <->  ( x  e.  A  |->  C )  Fn  A )
1613, 15sylibr 205 . . . 4  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  A. x  e.  A  C  e.  _V )
17 nfcsb1v 3088 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1817nfel1 2404 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  _V
19 csbeq1a 3064 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
2019eleq1d 2324 . . . . 5  |-  ( x  =  a  ->  ( C  e.  _V  <->  [_ a  /  x ]_ C  e.  _V ) )
2118, 20rcla4 2853 . . . 4  |-  ( a  e.  A  ->  ( A. x  e.  A  C  e.  _V  ->  [_ a  /  x ]_ C  e.  _V )
)
222, 16, 21sylc 58 . . 3  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e. 
_V )
23 nfcv 2394 . . . . 5  |-  F/_ a B
2423, 7, 9cbvmpt 4084 . . . 4  |-  ( x  e.  A  |->  B )  =  ( a  e.  A  |->  [_ a  /  x ]_ B )
2524a1i 12 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( x  e.  A  |->  B )  =  ( a  e.  A  |->  [_ a  /  x ]_ B ) )
26 nfcv 2394 . . . . 5  |-  F/_ a C
2726, 17, 19cbvmpt 4084 . . . 4  |-  ( x  e.  A  |->  C )  =  ( a  e.  A  |->  [_ a  /  x ]_ C )
2827a1i 12 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( x  e.  A  |->  C )  =  ( a  e.  A  |->  [_ a  /  x ]_ C ) )
291, 12, 22, 25, 28offval2 6029 . 2  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( (
x  e.  A  |->  B )  o F R ( x  e.  A  |->  C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) ) )
30 nfcv 2394 . . 3  |-  F/_ a
( B R C )
31 nfcv 2394 . . . 4  |-  F/_ x R
327, 31, 17nfov 5815 . . 3  |-  F/_ x
( [_ a  /  x ]_ B R [_ a  /  x ]_ C )
339, 19oveq12d 5810 . . 3  |-  ( x  =  a  ->  ( B R C )  =  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) )
3430, 32, 33cbvmpt 4084 . 2  |-  ( x  e.  A  |->  ( B R C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R
[_ a  /  x ]_ C ) )
3529, 34syl6eqr 2308 1  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( (
x  e.  A  |->  B )  o F R ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2518   _Vcvv 2763   [_csb 3056    e. cmpt 4051    Fn wfn 4668  (class class class)co 5792    o Fcof 6010
This theorem is referenced by:  mzpaddmpt  26187  mzpmulmpt  26188  mzpcompact2lem  26197
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012
  Copyright terms: Public domain W3C validator