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

Step	Hyp	Ref	Expression
1		predon 6608	. . . . . . 7 |- ( x e. On -> Pred ( _E , On , x ) = x )
2	1	reseq2d 5093	. . . . . 6 |- ( x e. On -> ( B |` Pred ( _E , On , x ) ) = ( B |` x ) )
3	2	fveq2d 5852	. . . . 5 |- ( x e. On -> ( G ` ( B |` Pred ( _E , On , x ) ) ) = ( G ` ( B |` x ) ) )
4	3	eqeq2d 2416	. . . 4 |- ( x e. On -> ( ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> ( B ` x ) = ( G ` ( B |` x ) ) ) )
5	4	ralbiia 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 )
10	8, 9	eqtri 2431	. . . 4 |- F = wrecs ( _E , On , G )
11	6, 7, 10	wfr3 7040	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) ) -> F = B )
12	5, 11	sylan2br 474	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> F = B )
13	12	eqcomd 2410	1 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )

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)