MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfr3ALT Structured version   Unicode version

Theorem tfr3ALT 7104
Description: Alternate proof of tfr3 7101 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
tfrALT.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr3ALT  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Distinct variable groups:    x, B    x, G    x, F

Proof of Theorem tfr3ALT
StepHypRef Expression
1 predon 6608 . . . . . . 7  |-  ( x  e.  On  ->  Pred (  _E  ,  On ,  x
)  =  x )
21reseq2d 5093 . . . . . 6  |-  ( x  e.  On  ->  ( B  |`  Pred (  _E  ,  On ,  x )
)  =  ( B  |`  x ) )
32fveq2d 5852 . . . . 5  |-  ( x  e.  On  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  x )
) )  =  ( G `  ( B  |`  x ) ) )
43eqeq2d 2416 . . . 4  |-  ( x  e.  On  ->  (
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  x ) ) ) )
54ralbiia 2833 . . 3  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
6 epweon 6600 . . . 4  |-  _E  We  On
7 epse 4805 . . . 4  |-  _E Se  On
8 tfrALT.1 . . . . 5  |-  F  = recs ( G )
9 df-recs 7074 . . . . 5  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
108, 9eqtri 2431 . . . 4  |-  F  = wrecs (  _E  ,  On ,  G )
116, 7, 10wfr3 7040 . . 3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) ) )  ->  F  =  B )
125, 11sylan2br 474 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  F  =  B )
1312eqcomd 2410 1  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    _E cep 4731    |` cres 4824   Predcpred 5365   Oncon0 5409    Fn wfn 5563   ` cfv 5568  wrecscwrecs 7011  recscrecs 7073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-wrecs 7012  df-recs 7074
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator