Theorem wfrlem3a 7417

Description: Lemma for well-founded recursion. Show membership in the class of acceptable functions. (Contributed by Scott Fenton, 31-Jul-2020.)

Hypotheses

Ref	Expression
wfrlem1.1	|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }
wfrlem3a.2	|- G e. _V

Assertion

Ref	Expression
wfrlem3a	|- ( G e. B <-> E. z ( G Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) )

Distinct variable groups:   A, f, w, x, y, z   f, F, w, x, y, z   R, f, w, x, y, z   f, G, x, y, z, w

Allowed substitution hints:   B( x, y, z, w, f)

Proof of Theorem wfrlem3a

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem3a.2	. 2 |- G e. _V
2		fneq1 5979	. . . 4 |- ( g = G -> ( g Fn z <-> G Fn z ) )
3		fveq1 6190	. . . . . 6 |- ( g = G -> ( g ` w ) = ( G ` w ) )
4		reseq1 5390	. . . . . . 7 |- ( g = G -> ( g |` Pred ( R , A , w ) ) = ( G |` Pred ( R , A , w ) ) )
5	4	fveq2d 6195	. . . . . 6 |- ( g = G -> ( F ` ( g |` Pred ( R , A , w ) ) ) = ( F ` ( G |` Pred ( R , A , w ) ) ) )
6	3, 5	eqeq12d 2637	. . . . 5 |- ( g = G -> ( ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) <-> ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) )
7	6	ralbidv 2986	. . . 4 |- ( g = G -> ( A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) <-> A. w e. z ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) )
8	2, 7	3anbi13d 1401	. . 3 |- ( g = G -> ( ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) <-> ( G Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) ) )
9	8	exbidv 1850	. 2 |- ( g = G -> ( E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) <-> E. z ( G Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) ) )
10		wfrlem1.1	. . 3 |- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }
11	10	wfrlem1 7414	. 2 |- B = { g | E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) }
12	1, 9, 11	elab2 3354	1 |- ( G e. B <-> E. z ( G Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   C_ wss 3574   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  wfrlem17  7431