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Theorem tfrlem3ag 5865
Description: Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)
Hypothesis
Ref Expression
tfrlem3.1  {  |  On  Fn  `  F `  |`  }
Assertion
Ref Expression
tfrlem3ag  G  _V  G  On  G  Fn  G `  F `  G  |`
Distinct variable groups:   ,,,,, F   , G,,,,
Allowed substitution hints:   (,,,,)

Proof of Theorem tfrlem3ag
StepHypRef Expression
1 fneq12 4935 . . . 4  G  Fn  G  Fn
2 simpll 481 . . . . . . 7  G  G
3 simpr 103 . . . . . . 7  G
42, 3fveq12d 5127 . . . . . 6  G  `
 G `
52, 3reseq12d 4556 . . . . . . 7  G  |`  G  |`
65fveq2d 5125 . . . . . 6  G  F `
 |`  F `  G  |`
74, 6eqeq12d 2051 . . . . 5  G  `  F `  |`  G `  F `  G  |`
8 simplr 482 . . . . 5  G
97, 8cbvraldva2 2531 . . . 4  G  `  F `  |`  G `  F `  G  |`
101, 9anbi12d 442 . . 3  G  Fn  `  F `  |`  G  Fn  G `  F `  G  |`
1110cbvrexdva 2534 . 2  G  On  Fn  `  F `  |`  On  G  Fn  G `  F `  G  |`
12 tfrlem3.1 . 2  {  |  On  Fn  `  F `  |`  }
1311, 12elab2g 2683 1  G  _V  G  On  G  Fn  G `  F `  G  |`
Colors of variables: wff set class
Syntax hints:   wi 4   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:  tfrlemisucaccv  5880
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