Theorem tfr3ALT 28939

Description: tfr3 7065 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (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, F   x, G   x, B

Proof of Theorem tfr3ALT

Step	Hyp	Ref	Expression
1		predon 28847	. . . . . 6 |- ( x e. On -> Pred ( _E , On , x ) = x )
2	1	reseq2d 5271	. . . . 5 |- ( x e. On -> ( B |` Pred ( _E , On , x ) ) = ( B |` x ) )
3	2	fveq2d 5868	. . . 4 |- ( x e. On -> ( G ` ( B |` Pred ( _E , On , x ) ) ) = ( G ` ( B |` x ) ) )
4	3	eqeq2d 2481	. . 3 |- ( x e. On -> ( ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> ( B ` x ) = ( G ` ( B |` x ) ) ) )
5	4	ralbiia 2894	. 2 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) )
6		epweon 6597	. . . 4 |- _E We On
7		epse 4862	. . . 4 |- _E Se On
8		tfrALT.1	. . . . 5 |- F = recs ( G )
9		tfrALTlem 28936	. . . . 5 |- recs ( G ) = wrecs ( _E , On , G )
10	8, 9	eqtri 2496	. . . 4 |- F = wrecs ( _E , On , G )
11	6, 7, 10	wfr3 28935	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) ) -> F = B )
12	11	eqcomd 2475	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) ) -> B = F )
13	5, 12	sylan2br 476	1 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   _E cep 4789   Oncon0 4878   |` cres 5001   Fn wfn 5581   ` cfv 5586  recscrecs 7038   Predcpred 28817  wrecscwrecs 28909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-recs 7039  df-pred 28818  df-wrecs 28910

This theorem is referenced by: (None)