Theorem ordtypelem2 8424

Description: Lemma for ordtype 8437. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	|- F = recs ( G )
ordtypelem.2	|- C = { w e. A | A. j e. ran h j R w }
ordtypelem.3	|- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
ordtypelem.5	|- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
ordtypelem.6	|- O = OrdIso ( R , A )
ordtypelem.7	|- ( ph -> R We A )
ordtypelem.8	|- ( ph -> R Se A )

Assertion

Ref	Expression
ordtypelem2	|- ( ph -> Ord T )

Distinct variable groups:   v, u, C   h, j, t, u, v, w, x, z, R   A, h, j, t, u, v, w, x, z   t, O, u, v, x   ph, t, x   h, F, j, t, u, v, w, x, z

Allowed substitution hints:   ph( z, w, v, u, h, j)   C( x, z, w, t, h, j)   T( x, z, w, v, u, t, h, j)   G( x, z, w, v, u, t, h, j)   O( z, w, h, j)

Proof of Theorem ordtypelem2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.5	. . . . . . . . . 10 |- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
2		ssrab2 3687	. . . . . . . . . 10 |- { x e. On | E. t e. A A. z e. ( F " x ) z R t } C_ On
3	1, 2	eqsstri 3635	. . . . . . . . 9 |- T C_ On
4	3	a1i 11	. . . . . . . 8 |- ( ph -> T C_ On )
5	4	sselda 3603	. . . . . . 7 |- ( ( ph /\ a e. T ) -> a e. On )
6		onss 6990	. . . . . . 7 |- ( a e. On -> a C_ On )
7	5, 6	syl 17	. . . . . 6 |- ( ( ph /\ a e. T ) -> a C_ On )
8		eloni 5733	. . . . . . . 8 |- ( a e. On -> Ord a )
9	5, 8	syl 17	. . . . . . 7 |- ( ( ph /\ a e. T ) -> Ord a )
10		imaeq2 5462	. . . . . . . . . . . 12 |- ( x = a -> ( F " x ) = ( F " a ) )
11	10	raleqdv 3144	. . . . . . . . . . 11 |- ( x = a -> ( A. z e. ( F " x ) z R t <-> A. z e. ( F " a ) z R t ) )
12	11	rexbidv 3052	. . . . . . . . . 10 |- ( x = a -> ( E. t e. A A. z e. ( F " x ) z R t <-> E. t e. A A. z e. ( F " a ) z R t ) )
13	12, 1	elrab2 3366	. . . . . . . . 9 |- ( a e. T <-> ( a e. On /\ E. t e. A A. z e. ( F " a ) z R t ) )
14	13	simprbi 480	. . . . . . . 8 |- ( a e. T -> E. t e. A A. z e. ( F " a ) z R t )
15	14	adantl 482	. . . . . . 7 |- ( ( ph /\ a e. T ) -> E. t e. A A. z e. ( F " a ) z R t )
16		ordelss 5739	. . . . . . . . 9 |- ( ( Ord a /\ x e. a ) -> x C_ a )
17		imass2 5501	. . . . . . . . 9 |- ( x C_ a -> ( F " x ) C_ ( F " a ) )
18		ssralv 3666	. . . . . . . . . 10 |- ( ( F " x ) C_ ( F " a ) -> ( A. z e. ( F " a ) z R t -> A. z e. ( F " x ) z R t ) )
19	18	reximdv 3016	. . . . . . . . 9 |- ( ( F " x ) C_ ( F " a ) -> ( E. t e. A A. z e. ( F " a ) z R t -> E. t e. A A. z e. ( F " x ) z R t ) )
20	16, 17, 19	3syl 18	. . . . . . . 8 |- ( ( Ord a /\ x e. a ) -> ( E. t e. A A. z e. ( F " a ) z R t -> E. t e. A A. z e. ( F " x ) z R t ) )
21	20	ralrimdva 2969	. . . . . . 7 |- ( Ord a -> ( E. t e. A A. z e. ( F " a ) z R t -> A. x e. a E. t e. A A. z e. ( F " x ) z R t ) )
22	9, 15, 21	sylc 65	. . . . . 6 |- ( ( ph /\ a e. T ) -> A. x e. a E. t e. A A. z e. ( F " x ) z R t )
23		ssrab 3680	. . . . . 6 |- ( a C_ { x e. On | E. t e. A A. z e. ( F " x ) z R t } <-> ( a C_ On /\ A. x e. a E. t e. A A. z e. ( F " x ) z R t ) )
24	7, 22, 23	sylanbrc 698	. . . . 5 |- ( ( ph /\ a e. T ) -> a C_ { x e. On | E. t e. A A. z e. ( F " x ) z R t } )
25	24, 1	syl6sseqr 3652	. . . 4 |- ( ( ph /\ a e. T ) -> a C_ T )
26	25	ralrimiva 2966	. . 3 |- ( ph -> A. a e. T a C_ T )
27		dftr3 4756	. . 3 |- ( Tr T <-> A. a e. T a C_ T )
28	26, 27	sylibr 224	. 2 |- ( ph -> Tr T )
29		ordon 6982	. . 3 |- Ord On
30		trssord 5740	. . 3 |- ( ( Tr T /\ T C_ On /\ Ord On ) -> Ord T )
31	3, 29, 30	mp3an23 1416	. 2 |- ( Tr T -> Ord T )
32	28, 31	syl 17	1 |- ( ph -> Ord T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   Tr wtr 4752   Se wse 5071   We wwe 5072   ran crn 5115   "cima 5117   Ord word 5722   Oncon0 5723   iota_crio 6610  recscrecs 7467  OrdIsocoi 8414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727

This theorem is referenced by:  ordtypelem5  8427  ordtypelem6  8428  ordtypelem7  8429  ordtypelem8  8430  ordtypelem9  8431