Theorem tfrlemi14d 6336

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 6321	. . 3 |- Ord dom recs ( F )
3		ordsson 4493	. . 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 6335	. . . . . . 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 5307	. . . . . . . . . . . . 13 |- ( w = z -> ( g Fn w <-> g Fn z ) )
11		raleq 2673	. . . . . . . . . . . . 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 2843	. . . . . . . . . . 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 2742	. . . . . . . . . . 11 |- g e. _V
16	1, 15	tfrlem3a 6313	. . . . . . . . . 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 6328	. . . . . . . 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 2742	. . . . . . . . . . . 12 |- z e. _V
20	5	tfrlem3-2d 6315	. . . . . . . . . . . . 13 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
21	20	simprd 114	. . . . . . . . . . . 12 |- ( ph -> ( F ` g ) e. _V )
22		opexg 4230	. . . . . . . . . . . 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 3623	. . . . . . . . . . 11 |- ( <. z , ( F ` g ) >. e. _V -> <. z , ( F ` g ) >. e. { <. z , ( F ` g ) >. } )
25		elun2 3305	. . . . . . . . . . 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 4834	. . . . . . . . . . 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 4829	. . . . . . . . . 10 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> dom h = dom ( g u. { <. z , ( F ` g ) >. } ) )
33	32	eleq2d 2247	. . . . . . . . 9 |- ( h = ( g u. { <. z , ( F ` g ) >. } ) -> ( z e. dom h <-> z e. dom ( g u. { <. z , ( F ` g ) >. } ) ) )
34	33	rspcev 2843	. . . . . . . 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 1898	. . . . . 6 |- ( ( ph /\ z e. On ) -> E. h e. A z e. dom h )
37		eliun 3892	. . . . . 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 3163	. . 3 |- ( ph -> On C_ U_ h e. A dom h )
41	1	recsfval 6318	. . . . 5 |- recs ( F ) = U. A
42	41	dmeqi 4830	. . . 4 |- dom recs ( F ) = dom U. A
43		dmuni 4839	. . . 4 |- dom U. A = U_ h e. A dom h
44	42, 43	eqtri 2198	. . 3 |- dom recs ( F ) = U_ h e. A dom h
45	40, 44	sseqtrrdi 3206	. 2 |- ( ph -> On C_ dom recs ( F ) )
46	4, 45	eqssd 3174	1 |- ( ph -> dom recs ( F ) = On )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2739   u. cun 3129   C_ wss 3131   {csn 3594   <.cop 3597   U.cuni 3811   U_ciun 3888   Ord word 4364   Oncon0 4365   dom cdm 4628   |` cres 4630   Fun wfun 5212   Fn wfn 5213   ` cfv 5218  recscrecs 6307

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308

This theorem is referenced by:  tfri1d  6338