Theorem tfrlemi14d 6238

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 6223	. . 3 |- Ord dom recs ( F )
3		ordsson 4416	. . 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 6237	. . . . . . 7 |- ( ( ph /\ z e. On ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) )
7	5	ad2antrr 480	. . . . . . . . 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 520	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> z e. On )
9		simprl 521	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> g Fn z )
10		fneq2 5220	. . . . . . . . . . . . 13 |- ( w = z -> ( g Fn w <-> g Fn z ) )
11		raleq 2629	. . . . . . . . . . . . 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 465	. . . . . . . . . . . 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 2793	. . . . . . . . . . 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 468	. . . . . . . . . 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 2692	. . . . . . . . . . 11 |- g e. _V
16	1, 15	tfrlem3a 6215	. . . . . . . . . 10 |- ( g e. A <-> E. w e. On ( g Fn w /\ A. u e. w ( g ` u ) = ( F ` ( g |` u ) ) ) )
17	14, 16	sylibr 133	. . . . . . . . 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 6230	. . . . . . . 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 2692	. . . . . . . . . . . 12 |- z e. _V
20	5	tfrlem3-2d 6217	. . . . . . . . . . . . 13 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
21	20	simprd 113	. . . . . . . . . . . 12 |- ( ph -> ( F ` g ) e. _V )
22		opexg 4158	. . . . . . . . . . . 12 |- ( ( z e. _V /\ ( F ` g ) e. _V ) -> <. z , ( F ` g ) >. e. _V )
23	19, 21, 22	sylancr 411	. . . . . . . . . . 11 |- ( ph -> <. z , ( F ` g ) >. e. _V )
24		snidg 3561	. . . . . . . . . . 11 |- ( <. z , ( F ` g ) >. e. _V -> <. z , ( F ` g ) >. e. { <. z , ( F ` g ) >. } )
25		elun2 3249	. . . . . . . . . . 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 480	. . . . . . . . 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 4752	. . . . . . . . . . 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 411	. . . . . . . . . 10 |- ( ph -> ( <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
30	29	ad2antrr 480	. . . . . . . . 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 4747	. . . . . . . . . 10 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> dom h = dom ( g u. { <. z , ( F ` g ) >. } ) )
33	32	eleq2d 2210	. . . . . . . . 9 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> ( z e. dom h <-> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
34	33	rspcev 2793	. . . . . . . 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 409	. . . . . . 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 1871	. . . . . 6 |- ( ( ph /\ z e. On ) -> E. h e. A z e. dom h )
37		eliun 3825	. . . . . 6 |- ( z e. U_ h e. A dom h <-> E. h e. A z e. dom h )
38	36, 37	sylibr 133	. . . . 5 |- ( ( ph /\ z e. On ) -> z e. U_ h e. A dom h )
39	38	ex 114	. . . 4 |- ( ph -> ( z e. On -> z e. U_ h e. A dom h ) )
40	39	ssrdv 3108	. . 3 |- ( ph -> On C_ U_ h e. A dom h )
41	1	recsfval 6220	. . . . 5 |- recs ( F ) = U. A
42	41	dmeqi 4748	. . . 4 |- dom recs ( F ) = dom U. A
43		dmuni 4757	. . . 4 |- dom U. A = U_ h e. A dom h
44	42, 43	eqtri 2161	. . 3 |- dom recs ( F ) = U_ h e. A dom h
45	40, 44	sseqtrrdi 3151	. 2 |- ( ph -> On C_ dom recs ( F ) )
46	4, 45	eqssd 3119	1 |- ( ph -> dom recs ( F ) = On )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   _Vcvv 2689   u. cun 3074   C_ wss 3076   {csn 3532   <.cop 3535   U.cuni 3744   U_ciun 3821   Ord word 4292   Oncon0 4293   dom cdm 4547   |` cres 4549   Fun wfun 5125   Fn wfn 5126   ` cfv 5131  recscrecs 6209

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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-recs 6210

This theorem is referenced by:  tfri1d  6240