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Theorem ofmpteq 26708
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 2430 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
54mptfng 5556 . . . . 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 3270 . . . . . 6  |-  F/_ x [_ a  /  x ]_ B
87nfel1 2576 . . . . 5  |-  F/ x [_ a  /  x ]_ B  e.  _V
9 csbeq1a 3246 . . . . . 6  |-  ( x  =  a  ->  B  =  [_ a  /  x ]_ B )
109eleq1d 2496 . . . . 5  |-  ( x  =  a  ->  ( B  e.  _V  <->  [_ a  /  x ]_ B  e.  _V ) )
118, 10rspc 3033 . . . 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 2430 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1514mptfng 5556 . . . . 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 3270 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1817nfel1 2576 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  _V
19 csbeq1a 3246 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
2019eleq1d 2496 . . . . 5  |-  ( x  =  a  ->  ( C  e.  _V  <->  [_ a  /  x ]_ C  e.  _V ) )
2118, 20rspc 3033 . . . 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 2566 . . . . 5  |-  F/_ a B
2423, 7, 9cbvmpt 4286 . . . 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 2566 . . . . 5  |-  F/_ a C
2726, 17, 19cbvmpt 4286 . . . 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 6308 . 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 2566 . . 3  |-  F/_ a
( B R C )
31 nfcv 2566 . . . 4  |-  F/_ x R
327, 31, 17nfov 6090 . . 3  |-  F/_ x
( [_ a  /  x ]_ B R [_ a  /  x ]_ C )
339, 19oveq12d 6085 . . 3  |-  ( x  =  a  ->  ( B R C )  =  ( [_ a  /  x ]_ B R [_ a  /  x ]_ C
) )
3430, 32, 33cbvmpt 4286 . 2  |-  ( x  e.  A  |->  ( B R C ) )  =  ( a  e.  A  |->  ( [_ a  /  x ]_ B R
[_ a  /  x ]_ C ) )
3529, 34syl6eqr 2480 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 1652    e. wcel 1725   A.wral 2692   _Vcvv 2943   [_csb 3238    e. cmpt 4253    Fn wfn 5435  (class class class)co 6067    o Fcof 6289
This theorem is referenced by:  mzpaddmpt  26730  mzpmulmpt  26731  mzpcompact2lem  26740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291
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