Theorem tfrlem7 6318

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 6317	. 2 |- Rel recs ( F )
3	1	recsfval 6316	. . . . . . . . 9 |- recs ( F ) = U. A
4	3	eleq2i 2244	. . . . . . . 8 |- ( <. x , u >. e. recs ( F ) <-> <. x , u >. e. U. A )
5		eluni 3813	. . . . . . . 8 |- ( <. x , u >. e. U. A <-> E. g ( <. x , u >. e. g /\ g e. A ) )
6	4, 5	bitri 184	. . . . . . 7 |- ( <. x , u >. e. recs ( F ) <-> E. g ( <. x , u >. e. g /\ g e. A ) )
7	3	eleq2i 2244	. . . . . . . 8 |- ( <. x , v >. e. recs ( F ) <-> <. x , v >. e. U. A )
8		eluni 3813	. . . . . . . 8 |- ( <. x , v >. e. U. A <-> E. h ( <. x , v >. e. h /\ h e. A ) )
9	7, 8	bitri 184	. . . . . . 7 |- ( <. x , v >. e. recs ( F ) <-> E. h ( <. x , v >. e. h /\ h e. A ) )
10	6, 9	anbi12i 460	. . . . . 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 1932	. . . . . 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 187	. . . . 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 4005	. . . . . . . . 9 |- ( x g u <-> <. x , u >. e. g )
14		df-br 4005	. . . . . . . . 9 |- ( x h v <-> <. x , v >. e. h )
15	13, 14	anbi12i 460	. . . . . . . 8 |- ( ( x g u /\ x h v ) <-> ( <. x , u >. e. g /\ <. x , v >. e. h ) )
16	1	tfrlem5 6315	. . . . . . . . 9 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )
17	16	impcom 125	. . . . . . . 8 |- ( ( ( x g u /\ x h v ) /\ ( g e. A /\ h e. A ) ) -> u = v )
18	15, 17	sylanbr 285	. . . . . . 7 |- ( ( ( <. x , u >. e. g /\ <. x , v >. e. h ) /\ ( g e. A /\ h e. A ) ) -> u = v )
19	18	an4s 588	. . . . . 6 |- ( ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) -> u = v )
20	19	exlimivv 1896	. . . . 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 121	. . . 4 |- ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
22	21	ax-gen 1449	. . 3 |- A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
23	22	gen2 1450	. 2 |- A. x A. u A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
24		dffun4 5228	. 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 942	1 |- Fun recs ( F )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   <.cop 3596   U.cuni 3810   class class class wbr 4004   Oncon0 4364   |` cres 4629   Rel wrel 4632   Fun wfun 5211   Fn wfn 5212   ` cfv 5217  recscrecs 6305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-recs 6306

This theorem is referenced by:  tfrlem9  6320  tfrfun  6321  tfrlemibfn  6329  tfrlemiubacc  6331  tfri1d  6336  rdgfun  6374