Theorem tfr3ALT 27715

Description: tfr3 6850 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 27623	. . . . . 6 |- ( x e. On -> Pred ( _E , On , x ) = x )
2	1	reseq2d 5105	. . . . 5 |- ( x e. On -> ( B |` Pred ( _E , On , x ) ) = ( B |` x ) )
3	2	fveq2d 5690	. . . 4 |- ( x e. On -> ( G ` ( B |` Pred ( _E , On , x ) ) ) = ( G ` ( B |` x ) ) )
4	3	eqeq2d 2449	. . 3 |- ( x e. On -> ( ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> ( B ` x ) = ( G ` ( B |` x ) ) ) )
5	4	ralbiia 2742	. 2 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) )
6		epweon 6390	. . . 4 |- _E We On
7		epse 4698	. . . 4 |- _E Se On
8		tfrALT.1	. . . . 5 |- F = recs ( G )
9		tfrALTlem 27712	. . . . 5 |- recs ( G ) = wrecs ( _E , On , G )
10	8, 9	eqtri 2458	. . . 4 |- F = wrecs ( _E , On , G )
11	6, 7, 10	wfr3 27711	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) ) -> F = B )
12	11	eqcomd 2443	. 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 1369   e. wcel 1756   A.wral 2710   _E cep 4625   Oncon0 4714   |` cres 4837   Fn wfn 5408   ` cfv 5413  recscrecs 6823   Predcpred 27593  wrecscwrecs 27685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6824  df-pred 27594  df-wrecs 27686

This theorem is referenced by: (None)