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Theorem evl1fval1lem 17890
Description: Lemma for evl1fval1 17891. (Contributed by AV, 11-Sep-2019.)
Hypotheses
Ref Expression
evl1fval1.q  |-  Q  =  (eval1 `  R )
evl1fval1.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
evl1fval1lem  |-  ( R  e.  _V  ->  Q  =  ( R evalSub1  B ) )

Proof of Theorem evl1fval1lem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
2 eqid 2454 . . 3  |-  ( 1o eval  R )  =  ( 1o eval  R )
3 evl1fval1.b . . 3  |-  B  =  ( Base `  R
)
41, 2, 3evl1fval 17888 . 2  |-  (eval1 `  R
)  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( 1o eval  R ) )
5 evl1fval1.q . . 3  |-  Q  =  (eval1 `  R )
65a1i 11 . 2  |-  ( R  e.  _V  ->  Q  =  (eval1 `  R ) )
7 fvex 5810 . . . . . 6  |-  ( Base `  R )  e.  _V
83, 7eqeltri 2538 . . . . 5  |-  B  e. 
_V
98pwid 3983 . . . 4  |-  B  e. 
~P B
10 eqid 2454 . . . . 5  |-  ( R evalSub1  B )  =  ( R evalSub1  B )
11 eqid 2454 . . . . 5  |-  ( 1o evalSub  R )  =  ( 1o evalSub  R )
1210, 11, 3evls1fval 17880 . . . 4  |-  ( ( R  e.  _V  /\  B  e.  ~P B
)  ->  ( R evalSub1  B
)  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  R ) `
 B ) ) )
139, 12mpan2 671 . . 3  |-  ( R  e.  _V  ->  ( R evalSub1  B )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  R ) `  B
) ) )
142, 3evlval 17735 . . . . 5  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
1514eqcomi 2467 . . . 4  |-  ( ( 1o evalSub  R ) `  B
)  =  ( 1o eval  R )
1615coeq2i 5109 . . 3  |-  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  R ) `
 B ) )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )  o.  ( 1o eval  R ) )
1713, 16syl6eq 2511 . 2  |-  ( R  e.  _V  ->  ( R evalSub1  B )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( 1o eval  R ) ) )
184, 6, 173eqtr4a 2521 1  |-  ( R  e.  _V  ->  Q  =  ( R evalSub1  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078   ~Pcpw 3969   {csn 3986    |-> cmpt 4459    X. cxp 4947    o. ccom 4953   ` cfv 5527  (class class class)co 6201   1oc1o 7024    ^m cmap 7325   Basecbs 14293   evalSub ces 17711   eval cevl 17712   evalSub1 ces1 17874  eval1ce1 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-evls 17713  df-evl 17714  df-evls1 17876  df-evl1 17877
This theorem is referenced by:  evl1fval1  17891
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