Theorem tfrlem7 6550

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 6549	. 2 |- Rel recs ( F )
3	1	recsfval 6548	. . . . . . . . 9 |- recs ( F ) = U. A
4	3	eleq2i 2301	. . . . . . . 8 |- ( <. x , u >. e. recs ( F ) <-> <. x , u >. e. U. A )
5		eluni 3919	. . . . . . . 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 2301	. . . . . . . 8 |- ( <. x , v >. e. recs ( F ) <-> <. x , v >. e. U. A )
8		eluni 3919	. . . . . . . 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 1988	. . . . . 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 4112	. . . . . . . . 9 |- ( x g u <-> <. x , u >. e. g )
14		df-br 4112	. . . . . . . . 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 6547	. . . . . . . . 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 592	. . . . . 6 |- ( ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) -> u = v )
20	19	exlimivv 1948	. . . . 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 1498	. . 3 |- A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
23	22	gen2 1499	. 2 |- A. x A. u A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
24		dffun4 5365	. 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 951	1 |- Fun recs ( F )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   <.cop 3694   U.cuni 3916   class class class wbr 4111   Oncon0 4486   |` cres 4753   Rel wrel 4756   Fun wfun 5348   Fn wfn 5349   ` cfv 5354  recscrecs 6537

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-recs 6538

This theorem is referenced by:  tfrlem9  6552  tfrfun  6553  tfrlemibfn  6561  tfrlemiubacc  6563  tfri1d  6568  rdgfun  6606