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Theorem seqomlem0 7106
Description: Lemma for seq𝜔. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
seqomlem0  |-  rec (
( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a F b )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  =  rec ( ( c  e. 
om ,  d  e. 
_V  |->  <. suc  c , 
( c F d ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )
Distinct variable groups:    F, a,
b, c, d    I,
a, b, c, d

Proof of Theorem seqomlem0
StepHypRef Expression
1 suceq 4932 . . . 4  |-  ( a  =  c  ->  suc  a  =  suc  c )
2 oveq1 6277 . . . 4  |-  ( a  =  c  ->  (
a F b )  =  ( c F b ) )
31, 2opeq12d 4211 . . 3  |-  ( a  =  c  ->  <. suc  a ,  ( a F b ) >.  =  <. suc  c ,  ( c F b ) >.
)
4 oveq2 6278 . . . 4  |-  ( b  =  d  ->  (
c F b )  =  ( c F d ) )
54opeq2d 4210 . . 3  |-  ( b  =  d  ->  <. suc  c ,  ( c F b ) >.  =  <. suc  c ,  ( c F d ) >.
)
63, 5cbvmpt2v 6350 . 2  |-  ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a F b ) >. )  =  ( c  e. 
om ,  d  e. 
_V  |->  <. suc  c , 
( c F d ) >. )
7 rdgeq1 7069 . 2  |-  ( ( a  e.  om , 
b  e.  _V  |->  <. suc  a ,  ( a F b ) >.
)  =  ( c  e.  om ,  d  e.  _V  |->  <. suc  c ,  ( c F d ) >. )  ->  rec ( ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a F b ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  =  rec (
( c  e.  om ,  d  e.  _V  |->  <. suc  c ,  ( c F d )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) )
86, 7ax-mp 5 1  |-  rec (
( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a F b )
>. ) ,  <. (/) ,  (  _I  `  I )
>. )  =  rec ( ( c  e. 
om ,  d  e. 
_V  |->  <. suc  c , 
( c F d ) >. ) ,  <. (/)
,  (  _I  `  I ) >. )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   _Vcvv 3106   (/)c0 3783   <.cop 4022    _I cid 4779   suc csuc 4869   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   omcom 6673   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-suc 4873  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068
This theorem is referenced by:  fnseqom  7112  seqom0g  7113  seqomsuc  7114
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