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Theorem rdgeq2 5899
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
rdgeq2  rec F ,  rec F ,

Proof of Theorem rdgeq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3084 . . . 4  u.  U_ 
dom  F `
 `  u.  U_  dom  F `  `
21mpteq2dv 3839 . . 3  _V  |->  u.  U_ 
dom  F `
 ` 
_V  |->  u.  U_  dom  F `  `
3 recseq 5862 . . 3  _V  |->  u.  U_  dom  F `
 ` 
_V  |->  u.  U_  dom  F `  `
recs  _V  |->  u.  U_  dom  F `
 ` recs  _V  |->  u.  U_  dom  F `
 `
42, 3syl 14 . 2 recs  _V  |->  u.  U_  dom  F `
 ` recs  _V  |->  u.  U_  dom  F `
 `
5 df-irdg 5897 . 2  rec F , recs 
_V  |->  u.  U_  dom  F `  `
6 df-irdg 5897 . 2  rec F , recs 
_V  |->  u.  U_  dom  F `  `
74, 5, 63eqtr4g 2094 1  rec F ,  rec F ,
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   _Vcvv 2551    u. cun 2909   U_ciun 3648    |-> cmpt 3809   dom cdm 4288   ` cfv 4845  recscrecs 5860   reccrdg 5896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-iota 4810  df-fv 4853  df-recs 5861  df-irdg 5897
This theorem is referenced by:  rdg0g  5915  oav  5973
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