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Theorem seqomlem0 6707
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 4647 . . . 4  |-  ( a  =  c  ->  suc  a  =  suc  c )
2 oveq1 6089 . . . 4  |-  ( a  =  c  ->  (
a F b )  =  ( c F b ) )
31, 2opeq12d 3993 . . 3  |-  ( a  =  c  ->  <. suc  a ,  ( a F b ) >.  =  <. suc  c ,  ( c F b ) >.
)
4 oveq2 6090 . . . 4  |-  ( b  =  d  ->  (
c F b )  =  ( c F d ) )
54opeq2d 3992 . . 3  |-  ( b  =  d  ->  <. suc  c ,  ( c F b ) >.  =  <. suc  c ,  ( c F d ) >.
)
63, 5cbvmpt2v 6153 . 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 6670 . 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 8 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 set class
Syntax hints:    = wceq 1653   _Vcvv 2957   (/)c0 3629   <.cop 3818    _I cid 4494   suc csuc 4584   omcom 4846   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   reccrdg 6668
This theorem is referenced by:  fnseqom  6713  seqom0g  6714  seqomsuc  6715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-suc 4588  df-iota 5419  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-recs 6634  df-rdg 6669
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