Theorem hsmexlem2 8807

Description: Lemma for hsmex 8812. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8950 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
hsmexlem.f	|- F = OrdIso ( _E , B )
hsmexlem.g	|- G = OrdIso ( _E , U_ a e. A B )

Assertion

Ref	Expression
hsmexlem2	|- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. (har ` ~P ( A X. C ) ) )

Distinct variable groups:   A, a   C, a

Allowed substitution hints:   B( a)   F( a)   G( a)

Proof of Theorem hsmexlem2

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

Step	Hyp	Ref	Expression
1		elpwi 4019	. . . . . 6 |- ( B e. ~P On -> B C_ On )
2	1	adantr 465	. . . . 5 |- ( ( B e. ~P On /\ dom F e. C ) -> B C_ On )
3	2	ralimi 2857	. . . 4 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> A. a e. A B C_ On )
4		iunss 4366	. . . 4 |- ( U_ a e. A B C_ On <-> A. a e. A B C_ On )
5	3, 4	sylibr 212	. . 3 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> U_ a e. A B C_ On )
6	5	3ad2ant3 1019	. 2 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> U_ a e. A B C_ On )
7		xpexg 6586	. . . 4 |- ( ( A e. _V /\ C e. On ) -> ( A X. C ) e. _V )
8	7	3adant3 1016	. . 3 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( A X. C ) e. _V )
9		nfv 1683	. . . . . . . . 9 |- F/ a C e. On
10		nfra1 2845	. . . . . . . . 9 |- F/ a A. a e. A ( B e. ~P On /\ dom F e. C )
11	9, 10	nfan 1875	. . . . . . . 8 |- F/ a ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) )
12		rsp 2830	. . . . . . . . 9 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> ( a e. A -> ( B e. ~P On /\ dom F e. C ) ) )
13		onelss 4920	. . . . . . . . . . . . . 14 |- ( C e. On -> ( dom F e. C -> dom F C_ C ) )
14	13	imp 429	. . . . . . . . . . . . 13 |- ( ( C e. On /\ dom F e. C ) -> dom F C_ C )
15	14	adantrl 715	. . . . . . . . . . . 12 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> dom F C_ C )
16	15	3adant3 1016	. . . . . . . . . . 11 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) /\ b e. B ) -> dom F C_ C )
17		hsmexlem.f	. . . . . . . . . . . . . . . . . . 19 |- F = OrdIso ( _E , B )
18	17	oismo 7965	. . . . . . . . . . . . . . . . . 18 |- ( B C_ On -> ( Smo F /\ ran F = B ) )
19	1, 18	syl 16	. . . . . . . . . . . . . . . . 17 |- ( B e. ~P On -> ( Smo F /\ ran F = B ) )
20	19	ad2antrl 727	. . . . . . . . . . . . . . . 16 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( Smo F /\ ran F = B ) )
21	20	simprd 463	. . . . . . . . . . . . . . 15 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ran F = B )
22	17	oif 7955	. . . . . . . . . . . . . . 15 |- F : dom F --> B
23	21, 22	jctil 537	. . . . . . . . . . . . . 14 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( F : dom F --> B /\ ran F = B ) )
24		dffo2 5799	. . . . . . . . . . . . . 14 |- ( F : dom F -onto-> B <-> ( F : dom F --> B /\ ran F = B ) )
25	23, 24	sylibr 212	. . . . . . . . . . . . 13 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> F : dom F -onto-> B )
26		dffo3 6036	. . . . . . . . . . . . . 14 |- ( F : dom F -onto-> B <-> ( F : dom F --> B /\ A. b e. B E. e e. dom F b = ( F ` e ) ) )
27	26	simprbi 464	. . . . . . . . . . . . 13 |- ( F : dom F -onto-> B -> A. b e. B E. e e. dom F b = ( F ` e ) )
28		rsp 2830	. . . . . . . . . . . . 13 |- ( A. b e. B E. e e. dom F b = ( F ` e ) -> ( b e. B -> E. e e. dom F b = ( F ` e ) ) )
29	25, 27, 28	3syl 20	. . . . . . . . . . . 12 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( b e. B -> E. e e. dom F b = ( F ` e ) ) )
30	29	3impia 1193	. . . . . . . . . . 11 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) /\ b e. B ) -> E. e e. dom F b = ( F ` e ) )
31		ssrexv 3565	. . . . . . . . . . 11 |- ( dom F C_ C -> ( E. e e. dom F b = ( F ` e ) -> E. e e. C b = ( F ` e ) ) )
32	16, 30, 31	sylc 60	. . . . . . . . . 10 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) /\ b e. B ) -> E. e e. C b = ( F ` e ) )
33	32	3exp 1195	. . . . . . . . 9 |- ( C e. On -> ( ( B e. ~P On /\ dom F e. C ) -> ( b e. B -> E. e e. C b = ( F ` e ) ) ) )
34	12, 33	sylan9r 658	. . . . . . . 8 |- ( ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( a e. A -> ( b e. B -> E. e e. C b = ( F ` e ) ) ) )
35	11, 34	reximdai 2933	. . . . . . 7 |- ( ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( E. a e. A b e. B -> E. a e. A E. e e. C b = ( F ` e ) ) )
36	35	3adant1 1014	. . . . . 6 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( E. a e. A b e. B -> E. a e. A E. e e. C b = ( F ` e ) ) )
37		nfv 1683	. . . . . . 7 |- F/ d E. e e. C b = ( F ` e )
38		nfcv 2629	. . . . . . . 8 |- F/_ a C
39		nfcv 2629	. . . . . . . . . . 11 |- F/_ a _E
40		nfcsb1v 3451	. . . . . . . . . . 11 |- F/_ a [_ d / a ]_ B
41	39, 40	nfoi 7939	. . . . . . . . . 10 |- F/_ aOrdIso ( _E , [_ d / a ]_ B )
42		nfcv 2629	. . . . . . . . . 10 |- F/_ a e
43	41, 42	nffv 5873	. . . . . . . . 9 |- F/_ a (OrdIso ( _E , [_ d / a ]_ B ) ` e )
44	43	nfeq2 2646	. . . . . . . 8 |- F/ a b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
45	38, 44	nfrex 2927	. . . . . . 7 |- F/ a E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
46		csbeq1a 3444	. . . . . . . . . . . 12 |- ( a = d -> B = [_ d / a ]_ B )
47		oieq2 7938	. . . . . . . . . . . 12 |- ( B = [_ d / a ]_ B -> OrdIso ( _E , B ) = OrdIso ( _E , [_ d / a ]_ B ) )
48	46, 47	syl 16	. . . . . . . . . . 11 |- ( a = d -> OrdIso ( _E , B ) = OrdIso ( _E , [_ d / a ]_ B ) )
49	17, 48	syl5eq 2520	. . . . . . . . . 10 |- ( a = d -> F = OrdIso ( _E , [_ d / a ]_ B ) )
50	49	fveq1d 5868	. . . . . . . . 9 |- ( a = d -> ( F ` e ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
51	50	eqeq2d 2481	. . . . . . . 8 |- ( a = d -> ( b = ( F ` e ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
52	51	rexbidv 2973	. . . . . . 7 |- ( a = d -> ( E. e e. C b = ( F ` e ) <-> E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
53	37, 45, 52	cbvrex 3085	. . . . . 6 |- ( E. a e. A E. e e. C b = ( F ` e ) <-> E. d e. A E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
54	36, 53	syl6ib 226	. . . . 5 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( E. a e. A b e. B -> E. d e. A E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
55		eliun 4330	. . . . 5 |- ( b e. U_ a e. A B <-> E. a e. A b e. B )
56		vex 3116	. . . . . . . . . . 11 |- d e. _V
57		vex 3116	. . . . . . . . . . 11 |- e e. _V
58	56, 57	op1std 6794	. . . . . . . . . 10 |- ( c = <. d , e >. -> ( 1st ` c ) = d )
59	58	csbeq1d 3442	. . . . . . . . 9 |- ( c = <. d , e >. -> [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B )
60		oieq2 7938	. . . . . . . . 9 |- ( [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B -> OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) = OrdIso ( _E , [_ d / a ]_ B ) )
61	59, 60	syl 16	. . . . . . . 8 |- ( c = <. d , e >. -> OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) = OrdIso ( _E , [_ d / a ]_ B ) )
62	56, 57	op2ndd 6795	. . . . . . . 8 |- ( c = <. d , e >. -> ( 2nd ` c ) = e )
63	61, 62	fveq12d 5872	. . . . . . 7 |- ( c = <. d , e >. -> (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
64	63	eqeq2d 2481	. . . . . 6 |- ( c = <. d , e >. -> ( b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
65	64	rexxp 5145	. . . . 5 |- ( E. c e. ( A X. C ) b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) <-> E. d e. A E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
66	54, 55, 65	3imtr4g 270	. . . 4 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( b e. U_ a e. A B -> E. c e. ( A X. C ) b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) ) )
67	66	imp 429	. . 3 |- ( ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) /\ b e. U_ a e. A B ) -> E. c e. ( A X. C ) b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) )
68	8, 67	wdomd 8007	. 2 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> U_ a e. A B ~<_* ( A X. C ) )
69		hsmexlem.g	. . 3 |- G = OrdIso ( _E , U_ a e. A B )
70	69	hsmexlem1 8806	. 2 |- ( ( U_ a e. A B C_ On /\ U_ a e. A B ~<_* ( A X. C ) ) -> dom G e. (har ` ~P ( A X. C ) ) )
71	6, 68, 70	syl2anc 661	1 |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. (har ` ~P ( A X. C ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   [_csb 3435   C_ wss 3476   ~Pcpw 4010   <.cop 4033   U_ciun 4325   class class class wbr 4447   _E cep 4789   Oncon0 4878   X. cxp 4997   dom cdm 4999   ran crn 5000   -->wf 5584   -onto->wfo 5586   ` cfv 5588   1stc1st 6782   2ndc2nd 6783   Smo wsmo 7016  OrdIsocoi 7934  harchar 7982   ~<_^* cwdom 7983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-1st 6784  df-2nd 6785  df-smo 7017  df-recs 7042  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-oi 7935  df-har 7984  df-wdom 7985

This theorem is referenced by:  hsmexlem3  8808