Theorem ordtypelem8 8430

Description: Lemma for ordtype 8437. (Contributed by Mario Carneiro, 25-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
ordtypelem8	|- ( ph -> O Isom _E , R ( dom O , ran O ) )

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 ordtypelem8

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . . . . 6 |- F = recs ( G )
2		ordtypelem.2	. . . . . 6 |- C = { w e. A | A. j e. ran h j R w }
3		ordtypelem.3	. . . . . 6 |- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
4		ordtypelem.5	. . . . . 6 |- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
5		ordtypelem.6	. . . . . 6 |- O = OrdIso ( R , A )
6		ordtypelem.7	. . . . . 6 |- ( ph -> R We A )
7		ordtypelem.8	. . . . . 6 |- ( ph -> R Se A )
8	1, 2, 3, 4, 5, 6, 7	ordtypelem4 8426	. . . . 5 |- ( ph -> O : ( T i^i dom F ) --> A )
9		fdm 6051	. . . . 5 |- ( O : ( T i^i dom F ) --> A -> dom O = ( T i^i dom F ) )
10	8, 9	syl 17	. . . 4 |- ( ph -> dom O = ( T i^i dom F ) )
11		inss1 3833	. . . . 5 |- ( T i^i dom F ) C_ T
12	1, 2, 3, 4, 5, 6, 7	ordtypelem2 8424	. . . . . 6 |- ( ph -> Ord T )
13		ordsson 6989	. . . . . 6 |- ( Ord T -> T C_ On )
14	12, 13	syl 17	. . . . 5 |- ( ph -> T C_ On )
15	11, 14	syl5ss 3614	. . . 4 |- ( ph -> ( T i^i dom F ) C_ On )
16	10, 15	eqsstrd 3639	. . 3 |- ( ph -> dom O C_ On )
17		epweon 6983	. . . 4 |- _E We On
18		weso 5105	. . . 4 |- ( _E We On -> _E Or On )
19	17, 18	ax-mp 5	. . 3 |- _E Or On
20		soss 5053	. . 3 |- ( dom O C_ On -> ( _E Or On -> _E Or dom O ) )
21	16, 19, 20	mpisyl 21	. 2 |- ( ph -> _E Or dom O )
22		frn 6053	. . . . 5 |- ( O : ( T i^i dom F ) --> A -> ran O C_ A )
23	8, 22	syl 17	. . . 4 |- ( ph -> ran O C_ A )
24		wess 5101	. . . 4 |- ( ran O C_ A -> ( R We A -> R We ran O ) )
25	23, 6, 24	sylc 65	. . 3 |- ( ph -> R We ran O )
26		weso 5105	. . 3 |- ( R We ran O -> R Or ran O )
27		sopo 5052	. . 3 |- ( R Or ran O -> R Po ran O )
28	25, 26, 27	3syl 18	. 2 |- ( ph -> R Po ran O )
29		ffun 6048	. . . 4 |- ( O : ( T i^i dom F ) --> A -> Fun O )
30	8, 29	syl 17	. . 3 |- ( ph -> Fun O )
31		funforn 6122	. . 3 |- ( Fun O <-> O : dom O -onto-> ran O )
32	30, 31	sylib 208	. 2 |- ( ph -> O : dom O -onto-> ran O )
33		epel 5032	. . . . 5 |- ( a _E b <-> a e. b )
34	1, 2, 3, 4, 5, 6, 7	ordtypelem6 8428	. . . . 5 |- ( ( ph /\ b e. dom O ) -> ( a e. b -> ( O ` a ) R ( O ` b ) ) )
35	33, 34	syl5bi 232	. . . 4 |- ( ( ph /\ b e. dom O ) -> ( a _E b -> ( O ` a ) R ( O ` b ) ) )
36	35	ralrimiva 2966	. . 3 |- ( ph -> A. b e. dom O ( a _E b -> ( O ` a ) R ( O ` b ) ) )
37	36	ralrimivw 2967	. 2 |- ( ph -> A. a e. dom O A. b e. dom O ( a _E b -> ( O ` a ) R ( O ` b ) ) )
38		soisoi 6578	. 2 |- ( ( ( _E Or dom O /\ R Po ran O ) /\ ( O : dom O -onto-> ran O /\ A. a e. dom O A. b e. dom O ( a _E b -> ( O ` a ) R ( O ` b ) ) ) ) -> O Isom _E , R ( dom O , ran O ) )
39	21, 28, 32, 37, 38	syl22anc 1327	1 |- ( ph -> O Isom _E , R ( dom O , ran O ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   Po wpo 5033   Or wor 5034   Se wse 5071   We wwe 5072   dom cdm 5114   ran crn 5115   "cima 5117   Ord word 5722   Oncon0 5723   Fun wfun 5882   -->wf 5884   -onto->wfo 5886   ` cfv 5888   Isom wiso 5889   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-pow 4843  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-oi 8415

This theorem is referenced by:  ordtypelem9  8431  ordtypelem10  8432  oiiso2  8436