Theorem tfr3ALT 27889

Description: tfr3 6967 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 27797	. . . . . 6 |- ( x e. On -> Pred ( _E , On , x ) = x )
2	1	reseq2d 5217	. . . . 5 |- ( x e. On -> ( B |` Pred ( _E , On , x ) ) = ( B |` x ) )
3	2	fveq2d 5802	. . . 4 |- ( x e. On -> ( G ` ( B |` Pred ( _E , On , x ) ) ) = ( G ` ( B |` x ) ) )
4	3	eqeq2d 2468	. . 3 |- ( x e. On -> ( ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> ( B ` x ) = ( G ` ( B |` x ) ) ) )
5	4	ralbiia 2837	. 2 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) )
6		epweon 6504	. . . 4 |- _E We On
7		epse 4810	. . . 4 |- _E Se On
8		tfrALT.1	. . . . 5 |- F = recs ( G )
9		tfrALTlem 27886	. . . . 5 |- recs ( G ) = wrecs ( _E , On , G )
10	8, 9	eqtri 2483	. . . 4 |- F = wrecs ( _E , On , G )
11	6, 7, 10	wfr3 27885	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) ) -> F = B )
12	11	eqcomd 2462	. 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 1370   e. wcel 1758   A.wral 2798   _E cep 4737   Oncon0 4826   |` cres 4949   Fn wfn 5520   ` cfv 5525  recscrecs 6940   Predcpred 27767  wrecscwrecs 27859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-recs 6941  df-pred 27768  df-wrecs 27860

This theorem is referenced by: (None)