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Theorem ofmpteq 26467
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 2387 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54mptfng 5510 . . . . 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 3226 . . . . . 6  |-  F/_ x [_ a  /  x ]_ B
87nfel1 2533 . . . . 5  |-  F/ x [_ a  /  x ]_ B  e.  _V
9 csbeq1a 3202 . . . . . 6  |-  ( x  =  a  ->  B  =  [_ a  /  x ]_ B )
109eleq1d 2453 . . . . 5  |-  ( x  =  a  ->  ( B  e.  _V  <->  [_ a  /  x ]_ B  e.  _V ) )
118, 10rspc 2989 . . . 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 2387 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1514mptfng 5510 . . . . 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 3226 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1817nfel1 2533 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  _V
19 csbeq1a 3202 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
2019eleq1d 2453 . . . . 5  |-  ( x  =  a  ->  ( C  e.  _V  <->  [_ a  /  x ]_ C  e.  _V ) )
2118, 20rspc 2989 . . . 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 2523 . . . . 5  |-  F/_ a B
2423, 7, 9cbvmpt 4240 . . . 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 2523 . . . . 5  |-  F/_ a C
2726, 17, 19cbvmpt 4240 . . . 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 6261 . 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 2523 . . 3  |-  F/_ a
( B R C )
31 nfcv 2523 . . . 4  |-  F/_ x R
327, 31, 17nfov 6043 . . 3  |-  F/_ x
( [_ a  /  x ]_ B R [_ a  /  x ]_ C )
339, 19oveq12d 6038 . . 3  |-  ( x  =  a  ->  ( B R C )  =  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) )
3430, 32, 33cbvmpt 4240 . 2  |-  ( x  e.  A  |->  ( B R C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R
[_ a  /  x ]_ C ) )
3529, 34syl6eqr 2437 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 2649   _Vcvv 2899   [_csb 3194    e. cmpt 4207    Fn wfn 5389  (class class class)co 6020    o Fcof 6242
This theorem is referenced by:  mzpaddmpt  26489  mzpmulmpt  26490  mzpcompact2lem  26499
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244
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