Theorem tfrlemi14d 6386

Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)

Hypotheses

Ref	Expression
tfrlemi14d.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlemi14d.2	|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )

Assertion

Ref	Expression
tfrlemi14d	|- ( ph -> dom recs ( F ) = On )

Distinct variable groups:   x, f, y, A   f, F, x, y   ph, f, y

Allowed substitution hint:   ph( x)

Proof of Theorem tfrlemi14d

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

Step	Hyp	Ref	Expression
1		tfrlemi14d.1	. . . 4 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem8 6371	. . 3 |- Ord dom recs ( F )
3		ordsson 4524	. . 3 |- ( Ord dom recs ( F ) -> dom recs ( F ) C_ On )
4	2, 3	mp1i 10	. 2 |- ( ph -> dom recs ( F ) C_ On )
5		tfrlemi14d.2	. . . . . . . 8 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
6	1, 5	tfrlemi1 6385	. . . . . . 7 |- ( ( ph /\ z e. On ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) )
7	5	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
8		simplr 528	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> z e. On )
9		simprl 529	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> g Fn z )
10		fneq2 5343	. . . . . . . . . . . . 13 |- ( w = z -> ( g Fn w <-> g Fn z ) )
11		raleq 2690	. . . . . . . . . . . . 13 |- ( w = z -> ( A. u e. w ( g ` u ) = ( F ` ( g |` u ) ) <-> A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) )
12	10, 11	anbi12d 473	. . . . . . . . . . . 12 |- ( w = z -> ( ( g Fn w /\ A. u e. w ( g ` u ) = ( F ` ( g |` u ) ) ) <-> ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
13	12	rspcev 2864	. . . . . . . . . . 11 |- ( ( z e. On /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> E. w e. On ( g Fn w /\ A. u e. w ( g ` u ) = ( F ` ( g |` u ) ) ) )
14	13	adantll 476	. . . . . . . . . 10 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> E. w e. On ( g Fn w /\ A. u e. w ( g ` u ) = ( F ` ( g |` u ) ) ) )
15		vex 2763	. . . . . . . . . . 11 |- g e. _V
16	1, 15	tfrlem3a 6363	. . . . . . . . . 10 |- ( g e. A <-> E. w e. On ( g Fn w /\ A. u e. w ( g ` u ) = ( F ` ( g |` u ) ) ) )
17	14, 16	sylibr 134	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> g e. A )
18	1, 7, 8, 9, 17	tfrlemisucaccv 6378	. . . . . . . 8 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> ( g u. { <. z , ( F ` g ) >. } ) e. A )
19		vex 2763	. . . . . . . . . . . 12 |- z e. _V
20	5	tfrlem3-2d 6365	. . . . . . . . . . . . 13 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
21	20	simprd 114	. . . . . . . . . . . 12 |- ( ph -> ( F ` g ) e. _V )
22		opexg 4257	. . . . . . . . . . . 12 |- ( ( z e. _V /\ ( F ` g ) e. _V ) -> <. z , ( F ` g ) >. e. _V )
23	19, 21, 22	sylancr 414	. . . . . . . . . . 11 |- ( ph -> <. z , ( F ` g ) >. e. _V )
24		snidg 3647	. . . . . . . . . . 11 |- ( <. z , ( F ` g ) >. e. _V -> <. z , ( F ` g ) >. e. { <. z , ( F ` g ) >. } )
25		elun2 3327	. . . . . . . . . . 11 |- ( <. z , ( F ` g ) >. e. { <. z , ( F ` g ) >. } -> <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) )
26	23, 24, 25	3syl 17	. . . . . . . . . 10 |- ( ph -> <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) )
27	26	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) )
28		opeldmg 4867	. . . . . . . . . . 11 |- ( ( z e. _V /\ ( F ` g ) e. _V ) -> ( <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
29	19, 21, 28	sylancr 414	. . . . . . . . . 10 |- ( ph -> ( <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
30	29	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> ( <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
31	27, 30	mpd 13	. . . . . . . 8 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) )
32		dmeq 4862	. . . . . . . . . 10 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> dom h = dom ( g u. { <. z , ( F ` g ) >. } ) )
33	32	eleq2d 2263	. . . . . . . . 9 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> ( z e. dom h <-> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
34	33	rspcev 2864	. . . . . . . 8 |- ( ( ( g u. { <. z , ( F ` g ) >. } ) e. A /\ z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) -> E. h e. A z e. dom h )
35	18, 31, 34	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> E. h e. A z e. dom h )
36	6, 35	exlimddv 1910	. . . . . 6 |- ( ( ph /\ z e. On ) -> E. h e. A z e. dom h )
37		eliun 3916	. . . . . 6 |- ( z e. U_ h e. A dom h <-> E. h e. A z e. dom h )
38	36, 37	sylibr 134	. . . . 5 |- ( ( ph /\ z e. On ) -> z e. U_ h e. A dom h )
39	38	ex 115	. . . 4 |- ( ph -> ( z e. On -> z e. U_ h e. A dom h ) )
40	39	ssrdv 3185	. . 3 |- ( ph -> On C_ U_ h e. A dom h )
41	1	recsfval 6368	. . . . 5 |- recs ( F ) = U. A
42	41	dmeqi 4863	. . . 4 |- dom recs ( F ) = dom U. A
43		dmuni 4872	. . . 4 |- dom U. A = U_ h e. A dom h
44	42, 43	eqtri 2214	. . 3 |- dom recs ( F ) = U_ h e. A dom h
45	40, 44	sseqtrrdi 3228	. 2 |- ( ph -> On C_ dom recs ( F ) )
46	4, 45	eqssd 3196	1 |- ( ph -> dom recs ( F ) = On )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   u. cun 3151   C_ wss 3153   {csn 3618   <.cop 3621   U.cuni 3835   U_ciun 3912   Ord word 4393   Oncon0 4394   dom cdm 4659   |` cres 4661   Fun wfun 5248   Fn wfn 5249   ` cfv 5254  recscrecs 6357

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358

This theorem is referenced by:  tfri1d  6388