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Theorem evl1fval1lem 18502
Description: Lemma for evl1fval1 18503. (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 2396 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
2 eqid 2396 . . 3  |-  ( 1o eval  R )  =  ( 1o eval  R )
3 evl1fval1.b . . 3  |-  B  =  ( Base `  R
)
41, 2, 3evl1fval 18500 . 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 5801 . . . . . 6  |-  ( Base `  R )  e.  _V
83, 7eqeltri 2480 . . . . 5  |-  B  e. 
_V
98pwid 3958 . . . 4  |-  B  e. 
~P B
10 eqid 2396 . . . . 5  |-  ( R evalSub1  B )  =  ( R evalSub1  B )
11 eqid 2396 . . . . 5  |-  ( 1o evalSub  R )  =  ( 1o evalSub  R )
1210, 11, 3evls1fval 18492 . . . 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 669 . . 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 18329 . . . . 5  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
1514eqcomi 2409 . . . 4  |-  ( ( 1o evalSub  R ) `  B
)  =  ( 1o eval  R )
1615coeq2i 5093 . . 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 2453 . 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 2463 1  |-  ( R  e.  _V  ->  Q  =  ( R evalSub1  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1836   _Vcvv 3051   ~Pcpw 3944   {csn 3961    |-> cmpt 4442    X. cxp 4928    o. ccom 4934   ` cfv 5513  (class class class)co 6218   1oc1o 7063    ^m cmap 7360   Basecbs 14657   evalSub ces 18305   eval cevl 18306   evalSub1 ces1 18486  eval1ce1 18487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-evls 18307  df-evl 18308  df-evls1 18488  df-evl1 18489
This theorem is referenced by:  evl1fval1  18503
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