Theorem tfrlem7 6285

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 6284	. 2 |- Rel recs ( F )
3	1	recsfval 6283	. . . . . . . . 9 |- recs ( F ) = U. A
4	3	eleq2i 2233	. . . . . . . 8 |- ( <. x , u >. e. recs ( F ) <-> <. x , u >. e. U. A )
5		eluni 3792	. . . . . . . 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 2233	. . . . . . . 8 |- ( <. x , v >. e. recs ( F ) <-> <. x , v >. e. U. A )
8		eluni 3792	. . . . . . . 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 456	. . . . . 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 1920	. . . . . 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 3983	. . . . . . . . 9 |- ( x g u <-> <. x , u >. e. g )
14		df-br 3983	. . . . . . . . 9 |- ( x h v <-> <. x , v >. e. h )
15	13, 14	anbi12i 456	. . . . . . . 8 |- ( ( x g u /\ x h v ) <-> ( <. x , u >. e. g /\ <. x , v >. e. h ) )
16	1	tfrlem5 6282	. . . . . . . . 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 578	. . . . . 6 |- ( ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) -> u = v )
20	19	exlimivv 1884	. . . . 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 1437	. . 3 |- A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
23	22	gen2 1438	. 2 |- A. x A. u A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
24		dffun4 5199	. 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 932	1 |- Fun recs ( F )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   <.cop 3579   U.cuni 3789   class class class wbr 3982   Oncon0 4341   |` cres 4606   Rel wrel 4609   Fun wfun 5182   Fn wfn 5183   ` cfv 5188  recscrecs 6272

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-recs 6273

This theorem is referenced by:  tfrlem9  6287  tfrfun  6288  tfrlemibfn  6296  tfrlemiubacc  6298  tfri1d  6303  rdgfun  6341