Theorem tfrlemi14d 6608

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 6593	. . 3 |- Ord dom recs ( F )
3		ordsson 4641	. . 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 6607	. . . . . . 7 |- ( ( ph /\ z e. On ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) )
7	5	ad2antrr 492	. . . . . . . . 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 533	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> z e. On )
9		simprl 535	. . . . . . . . 9 |- ( ( ( ph /\ z e. On ) /\ ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) -> g Fn z )
10		fneq2 5473	. . . . . . . . . . . . 13 |- ( w = z -> ( g Fn w <-> g Fn z ) )
11		raleq 2749	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 2929	. . . . . . . . . . 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 480	. . . . . . . . . 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 2824	. . . . . . . . . . 11 |- g e. _V
16	1, 15	tfrlem3a 6585	. . . . . . . . . 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 6600	. . . . . . . 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 2824	. . . . . . . . . . . 12 |- z e. _V
20	5	tfrlem3-2d 6587	. . . . . . . . . . . . 13 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
21	20	simprd 114	. . . . . . . . . . . 12 |- ( ph -> ( F ` g ) e. _V )
22		opexg 4370	. . . . . . . . . . . 12 |- ( ( z e. _V /\ ( F ` g ) e. _V ) -> <. z , ( F ` g ) >. e. _V )
23	19, 21, 22	sylancr 418	. . . . . . . . . . 11 |- ( ph -> <. z , ( F ` g ) >. e. _V )
24		snidg 3738	. . . . . . . . . . 11 |- ( <. z , ( F ` g ) >. e. _V -> <. z , ( F ` g ) >. e. { <. z , ( F ` g ) >. } )
25		elun2 3397	. . . . . . . . . . 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 492	. . . . . . . . 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 4988	. . . . . . . . . . 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 418	. . . . . . . . . 10 |- ( ph -> ( <. z , ( F ` g ) >. e. ( g u. { <. z , ( F ` g ) >. } ) -> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
30	29	ad2antrr 492	. . . . . . . . 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 4983	. . . . . . . . . 10 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> dom h = dom ( g u. { <. z , ( F ` g ) >. } ) )
33	32	eleq2d 2308	. . . . . . . . 9 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> ( z e. dom h <-> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
34	33	rspcev 2929	. . . . . . . 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 415	. . . . . . 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 1954	. . . . . 6 |- ( ( ph /\ z e. On ) -> E. h e. A z e. dom h )
37		eliun 4016	. . . . . 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 3254	. . 3 |- ( ph -> On C_ U_ h e. A dom h )
41	1	recsfval 6590	. . . . 5 |- recs ( F ) = U. A
42	41	dmeqi 4984	. . . 4 |- dom recs ( F ) = dom U. A
43		dmuni 4993	. . . 4 |- dom U. A = U_ h e. A dom h
44	42, 43	eqtri 2259	. . 3 |- dom recs ( F ) = U_ h e. A dom h
45	40, 44	sseqtrrdi 3297	. 2 |- ( ph -> On C_ dom recs ( F ) )
46	4, 45	eqssd 3265	1 |- ( ph -> dom recs ( F ) = On )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   u. cun 3218   C_ wss 3220   {csn 3709   <.cop 3712   U.cuni 3935   U_ciun 4012   Ord word 4509   Oncon0 4510   dom cdm 4776   |` cres 4778   Fun wfun 5374   Fn wfn 5375   ` cfv 5380  recscrecs 6579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-recs 6580

This theorem is used by:  tfri1d  6610