Theorem wfrrel 7420

Description: The well-founded recursion generator generates a relationship. (Contributed by Scott Fenton, 8-Jun-2018.)

Hypothesis

Ref	Expression
wfrlem6.1	|- F = wrecs ( R , A , G )

Assertion

Ref	Expression
wfrrel	|- Rel F

Proof of Theorem wfrrel

Dummy variables f g x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reluni 5241	. . 3 |- ( Rel U. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } <-> A. g e. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } Rel g )
2		eqid 2622	. . . . 5 |- { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } = { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) }
3	2	wfrlem2 7415	. . . 4 |- ( g e. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } -> Fun g )
4		funrel 5905	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 17	. . 3 |- ( g e. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } -> Rel g )
6	1, 5	mprgbir 2927	. 2 |- Rel U. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) }
7		wfrlem6.1	. . . 4 |- F = wrecs ( R , A , G )
8		df-wrecs 7407	. . . 4 |- wrecs ( R , A , G ) = U. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) }
9	7, 8	eqtri 2644	. . 3 |- F = U. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) }
10	9	releqi 5202	. 2 |- ( Rel F <-> Rel U. { 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 ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } )
11	6, 10	mpbir 221	1 |- Rel F

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   C_ wss 3574   U.cuni 4436   |` cres 5116   Rel wrel 5119   Predcpred 5679   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  wrecscwrecs 7406

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-iun 4522  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  df-wrecs 7407

This theorem is referenced by:  wfrfun  7425