Theorem tfrlem7 6214

Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem7	|- Fun recs ( F )

Distinct variable group:   x, f, y, F

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem7

Dummy variables g h u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . 3 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem6 6213	. 2 |- Rel recs ( F )
3	1	recsfval 6212	. . . . . . . . 9 |- recs ( F ) = U. A
4	3	eleq2i 2206	. . . . . . . 8 |- ( <. x , u >. e. recs ( F ) <-> <. x , u >. e. U. A )
5		eluni 3739	. . . . . . . 8 |- ( <. x , u >. e. U. A <-> E. g ( <. x , u >. e. g /\ g e. A ) )
6	4, 5	bitri 183	. . . . . . 7 |- ( <. x , u >. e. recs ( F ) <-> E. g ( <. x , u >. e. g /\ g e. A ) )
7	3	eleq2i 2206	. . . . . . . 8 |- ( <. x , v >. e. recs ( F ) <-> <. x , v >. e. U. A )
8		eluni 3739	. . . . . . . 8 |- ( <. x , v >. e. U. A <-> E. h ( <. x , v >. e. h /\ h e. A ) )
9	7, 8	bitri 183	. . . . . . 7 |- ( <. x , v >. e. recs ( F ) <-> E. h ( <. x , v >. e. h /\ h e. A ) )
10	6, 9	anbi12i 455	. . . . . 6 |- ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) <-> ( E. g ( <. x , u >. e. g /\ g e. A ) /\ E. h ( <. x , v >. e. h /\ h e. A ) ) )
11		eeanv 1904	. . . . . 6 |- ( E. g E. h ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) <-> ( E. g ( <. x , u >. e. g /\ g e. A ) /\ E. h ( <. x , v >. e. h /\ h e. A ) ) )
12	10, 11	bitr4i 186	. . . . 5 |- ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) <-> E. g E. h ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) )
13		df-br 3930	. . . . . . . . 9 |- ( x g u <-> <. x , u >. e. g )
14		df-br 3930	. . . . . . . . 9 |- ( x h v <-> <. x , v >. e. h )
15	13, 14	anbi12i 455	. . . . . . . 8 |- ( ( x g u /\ x h v ) <-> ( <. x , u >. e. g /\ <. x , v >. e. h ) )
16	1	tfrlem5 6211	. . . . . . . . 9 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )
17	16	impcom 124	. . . . . . . 8 |- ( ( ( x g u /\ x h v ) /\ ( g e. A /\ h e. A ) ) -> u = v )
18	15, 17	sylanbr 283	. . . . . . 7 |- ( ( ( <. x , u >. e. g /\ <. x , v >. e. h ) /\ ( g e. A /\ h e. A ) ) -> u = v )
19	18	an4s 577	. . . . . 6 |- ( ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) -> u = v )
20	19	exlimivv 1868	. . . . 5 |- ( E. g E. h ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) -> u = v )
21	12, 20	sylbi 120	. . . 4 |- ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
22	21	ax-gen 1425	. . 3 |- A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
23	22	gen2 1426	. 2 |- A. x A. u A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
24		dffun4 5134	. 2 |- ( Fun recs ( F ) <-> ( Rel recs ( F ) /\ A. x A. u A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v ) ) )
25	2, 23, 24	mpbir2an 926	1 |- Fun recs ( F )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   <.cop 3530   U.cuni 3736   class class class wbr 3929   Oncon0 4285   |` cres 4541   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118   ` cfv 5123  recscrecs 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-recs 6202

This theorem is referenced by:  tfrlem9  6216  tfrfun  6217  tfrlemibfn  6225  tfrlemiubacc  6227  tfri1d  6232  rdgfun  6270