Theorem nfwrecs 7409

Description: Bound-variable hypothesis builder for the well-founded recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.)

Hypotheses

Ref	Expression
nfwrecs.1	|- F/_ x R
nfwrecs.2	|- F/_ x A
nfwrecs.3	|- F/_ x F

Assertion

Ref	Expression
nfwrecs	|- F/_ xwrecs ( R , A , F )

Proof of Theorem nfwrecs

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

Step	Hyp	Ref	Expression
1		df-wrecs 7407	. 2 |- wrecs ( R , A , F ) = U. { f | E. y ( f Fn y /\ ( y C_ A /\ A. z e. y Pred ( R , A , z ) C_ y ) /\ A. z e. y ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) ) ) }
2		nfv 1843	. . . . . 6 |- F/ x f Fn y
3		nfcv 2764	. . . . . . . 8 |- F/_ x y
4		nfwrecs.2	. . . . . . . 8 |- F/_ x A
5	3, 4	nfss 3596	. . . . . . 7 |- F/ x y C_ A
6		nfwrecs.1	. . . . . . . . . 10 |- F/_ x R
7		nfcv 2764	. . . . . . . . . 10 |- F/_ x z
8	6, 4, 7	nfpred 5685	. . . . . . . . 9 |- F/_ x Pred ( R , A , z )
9	8, 3	nfss 3596	. . . . . . . 8 |- F/ x Pred ( R , A , z ) C_ y
10	3, 9	nfral 2945	. . . . . . 7 |- F/ x A. z e. y Pred ( R , A , z ) C_ y
11	5, 10	nfan 1828	. . . . . 6 |- F/ x ( y C_ A /\ A. z e. y Pred ( R , A , z ) C_ y )
12		nfwrecs.3	. . . . . . . . 9 |- F/_ x F
13		nfcv 2764	. . . . . . . . . 10 |- F/_ x f
14	13, 8	nfres 5398	. . . . . . . . 9 |- F/_ x ( f |` Pred ( R , A , z ) )
15	12, 14	nffv 6198	. . . . . . . 8 |- F/_ x ( F ` ( f |` Pred ( R , A , z ) ) )
16	15	nfeq2 2780	. . . . . . 7 |- F/ x ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) )
17	3, 16	nfral 2945	. . . . . 6 |- F/ x A. z e. y ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) )
18	2, 11, 17	nf3an 1831	. . . . 5 |- F/ x ( f Fn y /\ ( y C_ A /\ A. z e. y Pred ( R , A , z ) C_ y ) /\ A. z e. y ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) ) )
19	18	nfex 2154	. . . 4 |- F/ x E. y ( f Fn y /\ ( y C_ A /\ A. z e. y Pred ( R , A , z ) C_ y ) /\ A. z e. y ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) ) )
20	19	nfab 2769	. . 3 |- F/_ x { f | E. y ( f Fn y /\ ( y C_ A /\ A. z e. y Pred ( R , A , z ) C_ y ) /\ A. z e. y ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) ) ) }
21	20	nfuni 4442	. 2 |- F/_ x U. { f | E. y ( f Fn y /\ ( y C_ A /\ A. z e. y Pred ( R , A , z ) C_ y ) /\ A. z e. y ( f ` z ) = ( F ` ( f |` Pred ( R , A , z ) ) ) ) }
22	1, 21	nfcxfr 2762	1 |- F/_ xwrecs ( R , A , F )

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   {cab 2608   F/_wnfc 2751   A.wral 2912   C_ wss 3574   U.cuni 4436   |` cres 5116   Predcpred 5679   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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  nfrecs  7471