Theorem tfr3ALT 7498

Description: Alternate proof of tfr3 7495 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 6991	. . . . . . 7 |- ( x e. On -> Pred ( _E , On , x ) = x )
2	1	reseq2d 5396	. . . . . 6 |- ( x e. On -> ( B |` Pred ( _E , On , x ) ) = ( B |` x ) )
3	2	fveq2d 6195	. . . . 5 |- ( x e. On -> ( G ` ( B |` Pred ( _E , On , x ) ) ) = ( G ` ( B |` x ) ) )
4	3	eqeq2d 2632	. . . 4 |- ( x e. On -> ( ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> ( B ` x ) = ( G ` ( B |` x ) ) ) )
5	4	ralbiia 2979	. . 3 |- ( A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) <-> A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) )
6		epweon 6983	. . . 4 |- _E We On
7		epse 5097	. . . 4 |- _E Se On
8		tfrALT.1	. . . . 5 |- F = recs ( G )
9		df-recs 7468	. . . . 5 |- recs ( G ) = wrecs ( _E , On , G )
10	8, 9	eqtri 2644	. . . 4 |- F = wrecs ( _E , On , G )
11	6, 7, 10	wfr3 7435	. . 3 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` Pred ( _E , On , x ) ) ) ) -> F = B )
12	5, 11	sylan2br 493	. 2 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> F = B )
13	12	eqcomd 2628	1 |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _E cep 5028   |` cres 5116   Predcpred 5679   Oncon0 5723   Fn wfn 5883   ` cfv 5888  wrecscwrecs 7406  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by: (None)