Theorem tfrlemi14d 6312

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 6297	. . 3 |- Ord dom recs ( F )
3		ordsson 4476	. . 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 6311	. . . . . . 7 |- ( ( ph /\ z e. On ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) )
7	5	ad2antrr 485	. . . . . . . . 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 525	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> z e. On )
9		simprl 526	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> g Fn z )
10		fneq2 5287	. . . . . . . . . . . . 13 |- ( w = z -> ( g Fn w <-> g Fn z ) )
11		raleq 2665	. . . . . . . . . . . . 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 470	. . . . . . . . . . . 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 2834	. . . . . . . . . . 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 473	. . . . . . . . . 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 2733	. . . . . . . . . . 11 |- g e. _V
16	1, 15	tfrlem3a 6289	. . . . . . . . . 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 6304	. . . . . . . 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 2733	. . . . . . . . . . . 12 |- z e. _V
20	5	tfrlem3-2d 6291	. . . . . . . . . . . . 13 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
21	20	simprd 113	. . . . . . . . . . . 12 |- ( ph -> ( F ` g ) e. _V )
22		opexg 4213	. . . . . . . . . . . 12 |- ( ( z e. _V /\ ( F ` g ) e. _V ) -> <. z , ( F ` g ) >. e. _V )
23	19, 21, 22	sylancr 412	. . . . . . . . . . 11 |- ( ph -> <. z , ( F ` g ) >. e. _V )
24		snidg 3612	. . . . . . . . . . 11 |- ( <. z , ( F ` g ) >. e. _V -> <. z , ( F ` g ) >. e. { <. z , ( F ` g ) >. } )
25		elun2 3295	. . . . . . . . . . 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 485	. . . . . . . . 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 4816	. . . . . . . . . . 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 412	. . . . . . . . . 10 |- ( ph -> ( <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
30	29	ad2antrr 485	. . . . . . . . 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 4811	. . . . . . . . . 10 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> dom h = dom ( g u. { <. z , ( F ` g ) >. } ) )
33	32	eleq2d 2240	. . . . . . . . 9 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> ( z e. dom h <-> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
34	33	rspcev 2834	. . . . . . . 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 1891	. . . . . 6 |- ( ( ph /\ z e. On ) -> E. h e. A z e. dom h )
37		eliun 3877	. . . . . 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 3153	. . 3 |- ( ph -> On C_ U_ h e. A dom h )
41	1	recsfval 6294	. . . . 5 |- recs ( F ) = U. A
42	41	dmeqi 4812	. . . 4 |- dom recs ( F ) = dom U. A
43		dmuni 4821	. . . 4 |- dom U. A = U_ h e. A dom h
44	42, 43	eqtri 2191	. . 3 |- dom recs ( F ) = U_ h e. A dom h
45	40, 44	sseqtrrdi 3196	. 2 |- ( ph -> On C_ dom recs ( F ) )
46	4, 45	eqssd 3164	1 |- ( ph -> dom recs ( F ) = On )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   _Vcvv 2730   u. cun 3119   C_ wss 3121   {csn 3583   <.cop 3586   U.cuni 3796   U_ciun 3873   Ord word 4347   Oncon0 4348   dom cdm 4611   |` cres 4613   Fun wfun 5192   Fn wfn 5193   ` cfv 5198  recscrecs 6283

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-recs 6284

This theorem is referenced by:  tfri1d  6314