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Theorem ofmpteq 26460
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
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  A  e.  V )
2 simpr 448 . . . 4  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  a  e.  A )
3 simpl2 961 . . . . 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 2380 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54mptfng 5503 . . . . 5  |-  ( A. x  e.  A  B  e.  _V  <->  ( x  e.  A  |->  B )  Fn  A )
63, 5sylibr 204 . . . 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 3219 . . . . . 6  |-  F/_ x [_ a  /  x ]_ B
87nfel1 2526 . . . . 5  |-  F/ x [_ a  /  x ]_ B  e.  _V
9 csbeq1a 3195 . . . . . 6  |-  ( x  =  a  ->  B  =  [_ a  /  x ]_ B )
109eleq1d 2446 . . . . 5  |-  ( x  =  a  ->  ( B  e.  _V  <->  [_ a  /  x ]_ B  e.  _V ) )
118, 10rspc 2982 . . . 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 962 . . . . 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 2380 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1514mptfng 5503 . . . . 5  |-  ( A. x  e.  A  C  e.  _V  <->  ( x  e.  A  |->  C )  Fn  A )
1613, 15sylibr 204 . . . 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 3219 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1817nfel1 2526 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  _V
19 csbeq1a 3195 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
2019eleq1d 2446 . . . . 5  |-  ( x  =  a  ->  ( C  e.  _V  <->  [_ a  /  x ]_ C  e.  _V ) )
2118, 20rspc 2982 . . . 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 2516 . . . . 5  |-  F/_ a B
2423, 7, 9cbvmpt 4233 . . . 4  |-  ( x  e.  A  |->  B )  =  ( a  e.  A  |->  [_ a  /  x ]_ B )
2524a1i 11 . . 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 2516 . . . . 5  |-  F/_ a C
2726, 17, 19cbvmpt 4233 . . . 4  |-  ( x  e.  A  |->  C )  =  ( a  e.  A  |->  [_ a  /  x ]_ C )
2827a1i 11 . . 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 6254 . 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 2516 . . 3  |-  F/_ a
( B R C )
31 nfcv 2516 . . . 4  |-  F/_ x R
327, 31, 17nfov 6036 . . 3  |-  F/_ x
( [_ a  /  x ]_ B R [_ a  /  x ]_ C )
339, 19oveq12d 6031 . . 3  |-  ( x  =  a  ->  ( B R C )  =  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) )
3430, 32, 33cbvmpt 4233 . 2  |-  ( x  e.  A  |->  ( B R C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R
[_ a  /  x ]_ C ) )
3529, 34syl6eqr 2430 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 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892   [_csb 3187    e. cmpt 4200    Fn wfn 5382  (class class class)co 6013    o Fcof 6235
This theorem is referenced by:  mzpaddmpt  26482  mzpmulmpt  26483  mzpcompact2lem  26492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237
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