Theorem tfrlem7 6578

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 6577	. 2 |- Rel recs ( F )
3	1	recsfval 6576	. . . . . . . . 9 |- recs ( F ) = U. A
4	3	eleq2i 2305	. . . . . . . 8 |- ( <. x , u >. e. recs ( F ) <-> <. x , u >. e. U. A )
5		eluni 3933	. . . . . . . 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 2305	. . . . . . . 8 |- ( <. x , v >. e. recs ( F ) <-> <. x , v >. e. U. A )
8		eluni 3933	. . . . . . . 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 464	. . . . . 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 1992	. . . . . 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 4126	. . . . . . . . 9 |- ( x g u <-> <. x , u >. e. g )
14		df-br 4126	. . . . . . . . 9 |- ( x h v <-> <. x , v >. e. h )
15	13, 14	anbi12i 464	. . . . . . . 8 |- ( ( x g u /\ x h v ) <-> ( <. x , u >. e. g /\ <. x , v >. e. h ) )
16	1	tfrlem5 6575	. . . . . . . . 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 596	. . . . . 6 |- ( ( ( <. x , u >. e. g /\ g e. A ) /\ ( <. x , v >. e. h /\ h e. A ) ) -> u = v )
20	19	exlimivv 1952	. . . . 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 1502	. . 3 |- A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
23	22	gen2 1503	. 2 |- A. x A. u A. v ( ( <. x , u >. e. recs ( F ) /\ <. x , v >. e. recs ( F ) ) -> u = v )
24		dffun4 5383	. 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 955	1 |- Fun recs ( F )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   <.cop 3708   U.cuni 3930   class class class wbr 4125   Oncon0 4503   |` cres 4771   Rel wrel 4774   Fun wfun 5366   Fn wfn 5367   ` cfv 5372  recscrecs 6565

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-recs 6566

This theorem is referenced by:  tfrlem9  6580  tfrfun  6581  tfrlemibfn  6589  tfrlemiubacc  6591  tfri1d  6596  rdgfun  6634