Theorem ordtypelem10 8432

Description: Lemma for ordtype 8437. Using ax-rep 4771, exclude the possibility that O is a proper class and does not enumerate all of A. (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
ordtypelem10	|- ( ph -> O Isom _E , R ( dom O , A ) )

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 ordtypelem10

Dummy variables b c m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . 3 |- F = recs ( G )
2		ordtypelem.2	. . 3 |- C = { w e. A | A. j e. ran h j R w }
3		ordtypelem.3	. . 3 |- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
4		ordtypelem.5	. . 3 |- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
5		ordtypelem.6	. . 3 |- O = OrdIso ( R , A )
6		ordtypelem.7	. . 3 |- ( ph -> R We A )
7		ordtypelem.8	. . 3 |- ( ph -> R Se A )
8	1, 2, 3, 4, 5, 6, 7	ordtypelem8 8430	. 2 |- ( ph -> O Isom _E , R ( dom O , ran O ) )
9	1, 2, 3, 4, 5, 6, 7	ordtypelem4 8426	. . . . 5 |- ( ph -> O : ( T i^i dom F ) --> A )
10		frn 6053	. . . . 5 |- ( O : ( T i^i dom F ) --> A -> ran O C_ A )
11	9, 10	syl 17	. . . 4 |- ( ph -> ran O C_ A )
12		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> b e. A )
13	6	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> R We A )
14	7	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> R Se A )
15	1, 2, 3, 4, 5, 13, 14	ordtypelem8 8430	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> O Isom _E , R ( dom O , ran O ) )
16		isof1o 6573	. . . . . . . . . . . . 13 |- ( O Isom _E , R ( dom O , ran O ) -> O : dom O -1-1-onto-> ran O )
17		f1of 6137	. . . . . . . . . . . . 13 |- ( O : dom O -1-1-onto-> ran O -> O : dom O --> ran O )
18	15, 16, 17	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> O : dom O --> ran O )
19		f1of1 6136	. . . . . . . . . . . . . 14 |- ( O : dom O -1-1-onto-> ran O -> O : dom O -1-1-> ran O )
20	15, 16, 19	3syl 18	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> O : dom O -1-1-> ran O )
21		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( b e. A /\ -. b e. ran O ) -> b e. A )
22		seex 5077	. . . . . . . . . . . . . . 15 |- ( ( R Se A /\ b e. A ) -> { c e. A | c R b } e. _V )
23	7, 21, 22	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> { c e. A | c R b } e. _V )
24	11	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> ran O C_ A )
25		rexnal 2995	. . . . . . . . . . . . . . . . . . 19 |- ( E. m e. dom O -. ( O ` m ) R b <-> -. A. m e. dom O ( O ` m ) R b )
26	1, 2, 3, 4, 5, 6, 7	ordtypelem7 8429	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ b e. A ) /\ m e. dom O ) -> ( ( O ` m ) R b \/ b e. ran O ) )
27	26	ord 392	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ b e. A ) /\ m e. dom O ) -> ( -. ( O ` m ) R b -> b e. ran O ) )
28	27	rexlimdva 3031	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ b e. A ) -> ( E. m e. dom O -. ( O ` m ) R b -> b e. ran O ) )
29	25, 28	syl5bir 233	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ b e. A ) -> ( -. A. m e. dom O ( O ` m ) R b -> b e. ran O ) )
30	29	con1d 139	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ b e. A ) -> ( -. b e. ran O -> A. m e. dom O ( O ` m ) R b ) )
31	30	impr 649	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> A. m e. dom O ( O ` m ) R b )
32		ffun 6048	. . . . . . . . . . . . . . . . . . . 20 |- ( O : ( T i^i dom F ) --> A -> Fun O )
33	9, 32	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> Fun O )
34		funfn 5918	. . . . . . . . . . . . . . . . . . 19 |- ( Fun O <-> O Fn dom O )
35	33, 34	sylib 208	. . . . . . . . . . . . . . . . . 18 |- ( ph -> O Fn dom O )
36	35	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> O Fn dom O )
37		breq1 4656	. . . . . . . . . . . . . . . . . 18 |- ( c = ( O ` m ) -> ( c R b <-> ( O ` m ) R b ) )
38	37	ralrn 6362	. . . . . . . . . . . . . . . . 17 |- ( O Fn dom O -> ( A. c e. ran O c R b <-> A. m e. dom O ( O ` m ) R b ) )
39	36, 38	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> ( A. c e. ran O c R b <-> A. m e. dom O ( O ` m ) R b ) )
40	31, 39	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> A. c e. ran O c R b )
41		ssrab 3680	. . . . . . . . . . . . . . 15 |- ( ran O C_ { c e. A | c R b } <-> ( ran O C_ A /\ A. c e. ran O c R b ) )
42	24, 40, 41	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> ran O C_ { c e. A | c R b } )
43	23, 42	ssexd 4805	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> ran O e. _V )
44		f1dmex 7136	. . . . . . . . . . . . 13 |- ( ( O : dom O -1-1-> ran O /\ ran O e. _V ) -> dom O e. _V )
45	20, 43, 44	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> dom O e. _V )
46		fex 6490	. . . . . . . . . . . 12 |- ( ( O : dom O --> ran O /\ dom O e. _V ) -> O e. _V )
47	18, 45, 46	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> O e. _V )
48	1, 2, 3, 4, 5, 13, 14, 47	ordtypelem9 8431	. . . . . . . . . 10 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> O Isom _E , R ( dom O , A ) )
49		isof1o 6573	. . . . . . . . . 10 |- ( O Isom _E , R ( dom O , A ) -> O : dom O -1-1-onto-> A )
50		f1ofo 6144	. . . . . . . . . 10 |- ( O : dom O -1-1-onto-> A -> O : dom O -onto-> A )
51		forn 6118	. . . . . . . . . 10 |- ( O : dom O -onto-> A -> ran O = A )
52	48, 49, 50, 51	4syl 19	. . . . . . . . 9 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> ran O = A )
53	12, 52	eleqtrrd 2704	. . . . . . . 8 |- ( ( ph /\ ( b e. A /\ -. b e. ran O ) ) -> b e. ran O )
54	53	expr 643	. . . . . . 7 |- ( ( ph /\ b e. A ) -> ( -. b e. ran O -> b e. ran O ) )
55	54	pm2.18d 124	. . . . . 6 |- ( ( ph /\ b e. A ) -> b e. ran O )
56	55	ex 450	. . . . 5 |- ( ph -> ( b e. A -> b e. ran O ) )
57	56	ssrdv 3609	. . . 4 |- ( ph -> A C_ ran O )
58	11, 57	eqssd 3620	. . 3 |- ( ph -> ran O = A )
59		isoeq5 6571	. . 3 |- ( ran O = A -> ( O Isom _E , R ( dom O , ran O ) <-> O Isom _E , R ( dom O , A ) ) )
60	58, 59	syl 17	. 2 |- ( ph -> ( O Isom _E , R ( dom O , ran O ) <-> O Isom _E , R ( dom O , A ) ) )
61	8, 60	mpbid 222	1 |- ( ph -> O Isom _E , R ( dom O , A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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   Se wse 5071   We wwe 5072   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` 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-rep 4771  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:  ordtype  8437