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Theorem ofmpteq 6338
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 )  oF 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 988 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  A  e.  V )
2 simpr 461 . . . 4  |-  ( ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  /\  a  e.  A )  ->  a  e.  A )
3 simpl2 992 . . . . 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 2443 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54mptfng 5536 . . . . 5  |-  ( A. x  e.  A  B  e.  _V  <->  ( x  e.  A  |->  B )  Fn  A )
63, 5sylibr 212 . . . 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 3304 . . . . . 6  |-  F/_ x [_ a  /  x ]_ B
87nfel1 2589 . . . . 5  |-  F/ x [_ a  /  x ]_ B  e.  _V
9 csbeq1a 3297 . . . . . 6  |-  ( x  =  a  ->  B  =  [_ a  /  x ]_ B )
109eleq1d 2509 . . . . 5  |-  ( x  =  a  ->  ( B  e.  _V  <->  [_ a  /  x ]_ B  e.  _V ) )
118, 10rspc 3067 . . . 4  |-  ( a  e.  A  ->  ( A. x  e.  A  B  e.  _V  ->  [_ a  /  x ]_ B  e.  _V )
)
122, 6, 11sylc 60 . . 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 993 . . . . 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 2443 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1514mptfng 5536 . . . . 5  |-  ( A. x  e.  A  C  e.  _V  <->  ( x  e.  A  |->  C )  Fn  A )
1613, 15sylibr 212 . . . 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 3304 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1817nfel1 2589 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  _V
19 csbeq1a 3297 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
2019eleq1d 2509 . . . . 5  |-  ( x  =  a  ->  ( C  e.  _V  <->  [_ a  /  x ]_ C  e.  _V ) )
2118, 20rspc 3067 . . . 4  |-  ( a  e.  A  ->  ( A. x  e.  A  C  e.  _V  ->  [_ a  /  x ]_ C  e.  _V )
)
222, 16, 21sylc 60 . . 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 2579 . . . . 5  |-  F/_ a B
2423, 7, 9cbvmpt 4382 . . . 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 2579 . . . . 5  |-  F/_ a C
2726, 17, 19cbvmpt 4382 . . . 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 6336 . 2  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( (
x  e.  A  |->  B )  oF R ( x  e.  A  |->  C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) ) )
30 nfcv 2579 . . 3  |-  F/_ a
( B R C )
31 nfcv 2579 . . . 4  |-  F/_ x R
327, 31, 17nfov 6114 . . 3  |-  F/_ x
( [_ a  /  x ]_ B R [_ a  /  x ]_ C )
339, 19oveq12d 6109 . . 3  |-  ( x  =  a  ->  ( B R C )  =  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) )
3430, 32, 33cbvmpt 4382 . 2  |-  ( x  e.  A  |->  ( B R C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R
[_ a  /  x ]_ C ) )
3529, 34syl6eqr 2493 1  |-  ( ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A
)  ->  ( (
x  e.  A  |->  B )  oF R ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   [_csb 3288    e. cmpt 4350    Fn wfn 5413  (class class class)co 6091    oFcof 6318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320
This theorem is referenced by:  mdetrlin  18409  mzpaddmpt  29077  mzpmulmpt  29078  mzpcompact2lem  29088
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