Theorem wfrrel 7046

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 4959	. . 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 2453	. . . . 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 7041	. . . 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 5602	. . . 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 2754	. 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 7033	. . . 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 2475	. . 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 4921	. 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 213	1 |- Rel F

Syntax hints:   /\ wa 371   /\ w3a 986   = wceq 1446   E.wex 1665   e. wcel 1889   {cab 2439   A.wral 2739   C_ wss 3406   U.cuni 4201   |` cres 4839   Rel wrel 4842   Predcpred 5382   Fun wfun 5579   Fn wfn 5580   ` cfv 5585  wrecscwrecs 7032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-wrecs 7033

This theorem is referenced by:  wfrfun  7051