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Theorem tfrlem3a 5866
Description: Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)
Hypotheses
Ref Expression
tfrlem3.1  {  |  On  Fn  `  F `  |`  }
tfrlem3.2  G 
_V
Assertion
Ref Expression
tfrlem3a  G  On  G  Fn  G `  F `  G  |`
Distinct variable groups:   ,,,,, F   , G,,,,
Allowed substitution hints:   (,,,,)

Proof of Theorem tfrlem3a
StepHypRef Expression
1 tfrlem3.2 . 2  G 
_V
2 fneq12 4935 . . . 4  G  Fn  G  Fn
3 simpll 481 . . . . . . 7  G  G
4 simpr 103 . . . . . . 7  G
53, 4fveq12d 5127 . . . . . 6  G  `
 G `
63, 4reseq12d 4556 . . . . . . 7  G  |`  G  |`
76fveq2d 5125 . . . . . 6  G  F `
 |`  F `  G  |`
85, 7eqeq12d 2051 . . . . 5  G  `  F `  |`  G `  F `  G  |`
9 simpr 103 . . . . . 6  G
109adantr 261 . . . . 5  G
118, 10cbvraldva2 2531 . . . 4  G  `  F `  |`  G `  F `  G  |`
122, 11anbi12d 442 . . 3  G  Fn  `  F `  |`  G  Fn  G `  F `  G  |`
1312cbvrexdva 2534 . 2  G  On  Fn  `  F `  |`  On  G  Fn  G `  F `  G  |`
14 tfrlem3.1 . 2  {  |  On  Fn  `  F `  |`  }
151, 13, 14elab2 2684 1  G  On  G  Fn  G `  F `  G  |`
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   _Vcvv 2551   Oncon0 4066    |` cres 4290    Fn wfn 4840   ` cfv 4845
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-bndl 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-3an 886  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-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlem3  5867  tfrlem5  5871  tfrlemisucaccv  5880  tfrlemibxssdm  5882  tfrlemi14d  5888  tfrexlem  5889
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