Theorem hsmexlem2 8857

Description: Lemma for hsmex 8862. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9000 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 3960	. . . . . 6 |- ( B e. ~P On -> B C_ On )
2	1	adantr 467	. . . . 5 |- ( ( B e. ~P On /\ dom F e. C ) -> B C_ On )
3	2	ralimi 2781	. . . 4 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> A. a e. A B C_ On )
4		iunss 4319	. . . 4 |- ( U_ a e. A B C_ On <-> A. a e. A B C_ On )
5	3, 4	sylibr 216	. . 3 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> U_ a e. A B C_ On )
6	5	3ad2ant3 1031	. 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 6593	. . . 4 |- ( ( A e. _V /\ C e. On ) -> ( A X. C ) e. _V )
8	7	3adant3 1028	. . 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 1761	. . . . . . . . 9 |- F/ a C e. On
10		nfra1 2769	. . . . . . . . 9 |- F/ a A. a e. A ( B e. ~P On /\ dom F e. C )
11	9, 10	nfan 2011	. . . . . . . 8 |- F/ a ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) )
12		rsp 2754	. . . . . . . . 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 5465	. . . . . . . . . . . . . 14 |- ( C e. On -> ( dom F e. C -> dom F C_ C ) )
14	13	imp 431	. . . . . . . . . . . . 13 |- ( ( C e. On /\ dom F e. C ) -> dom F C_ C )
15	14	adantrl 722	. . . . . . . . . . . 12 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> dom F C_ C )
16	15	3adant3 1028	. . . . . . . . . . 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 8055	. . . . . . . . . . . . . . . . . 18 |- ( B C_ On -> ( Smo F /\ ran F = B ) )
19	1, 18	syl 17	. . . . . . . . . . . . . . . . 17 |- ( B e. ~P On -> ( Smo F /\ ran F = B ) )
20	19	ad2antrl 734	. . . . . . . . . . . . . . . 16 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( Smo F /\ ran F = B ) )
21	20	simprd 465	. . . . . . . . . . . . . . 15 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ran F = B )
22	17	oif 8045	. . . . . . . . . . . . . . 15 |- F : dom F --> B
23	21, 22	jctil 540	. . . . . . . . . . . . . 14 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( F : dom F --> B /\ ran F = B ) )
24		dffo2 5797	. . . . . . . . . . . . . 14 |- ( F : dom F -onto-> B <-> ( F : dom F --> B /\ ran F = B ) )
25	23, 24	sylibr 216	. . . . . . . . . . . . 13 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> F : dom F -onto-> B )
26		dffo3 6037	. . . . . . . . . . . . . 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 466	. . . . . . . . . . . . 13 |- ( F : dom F -onto-> B -> A. b e. B E. e e. dom F b = ( F ` e ) )
28		rsp 2754	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 1205	. . . . . . . . . . 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 3494	. . . . . . . . . . 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 62	. . . . . . . . . 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 1207	. . . . . . . . 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 664	. . . . . . . 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 2856	. . . . . . 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 1026	. . . . . 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 1761	. . . . . . 7 |- F/ d E. e e. C b = ( F ` e )
38		nfcv 2592	. . . . . . . 8 |- F/_ a C
39		nfcv 2592	. . . . . . . . . . 11 |- F/_ a _E
40		nfcsb1v 3379	. . . . . . . . . . 11 |- F/_ a [_ d / a ]_ B
41	39, 40	nfoi 8029	. . . . . . . . . 10 |- F/_ aOrdIso ( _E , [_ d / a ]_ B )
42		nfcv 2592	. . . . . . . . . 10 |- F/_ a e
43	41, 42	nffv 5872	. . . . . . . . 9 |- F/_ a (OrdIso ( _E , [_ d / a ]_ B ) ` e )
44	43	nfeq2 2607	. . . . . . . 8 |- F/ a b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
45	38, 44	nfrex 2850	. . . . . . 7 |- F/ a E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
46		csbeq1a 3372	. . . . . . . . . . . 12 |- ( a = d -> B = [_ d / a ]_ B )
47		oieq2 8028	. . . . . . . . . . . 12 |- ( B = [_ d / a ]_ B -> OrdIso ( _E , B ) = OrdIso ( _E , [_ d / a ]_ B ) )
48	46, 47	syl 17	. . . . . . . . . . 11 |- ( a = d -> OrdIso ( _E , B ) = OrdIso ( _E , [_ d / a ]_ B ) )
49	17, 48	syl5eq 2497	. . . . . . . . . 10 |- ( a = d -> F = OrdIso ( _E , [_ d / a ]_ B ) )
50	49	fveq1d 5867	. . . . . . . . 9 |- ( a = d -> ( F ` e ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
51	50	eqeq2d 2461	. . . . . . . 8 |- ( a = d -> ( b = ( F ` e ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
52	51	rexbidv 2901	. . . . . . 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 3016	. . . . . 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 230	. . . . 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 4283	. . . . 5 |- ( b e. U_ a e. A B <-> E. a e. A b e. B )
56		vex 3048	. . . . . . . . . . 11 |- d e. _V
57		vex 3048	. . . . . . . . . . 11 |- e e. _V
58	56, 57	op1std 6803	. . . . . . . . . 10 |- ( c = <. d , e >. -> ( 1st ` c ) = d )
59	58	csbeq1d 3370	. . . . . . . . 9 |- ( c = <. d , e >. -> [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B )
60		oieq2 8028	. . . . . . . . 9 |- ( [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B -> OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) = OrdIso ( _E , [_ d / a ]_ B ) )
61	59, 60	syl 17	. . . . . . . 8 |- ( c = <. d , e >. -> OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) = OrdIso ( _E , [_ d / a ]_ B ) )
62	56, 57	op2ndd 6804	. . . . . . . 8 |- ( c = <. d , e >. -> ( 2nd ` c ) = e )
63	61, 62	fveq12d 5871	. . . . . . 7 |- ( c = <. d , e >. -> (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
64	63	eqeq2d 2461	. . . . . 6 |- ( c = <. d , e >. -> ( b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
65	64	rexxp 4977	. . . . 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 274	. . . 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 431	. . 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 8096	. 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 8856	. 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 667	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 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045   [_csb 3363   C_ wss 3404   ~Pcpw 3951   <.cop 3974   U_ciun 4278   class class class wbr 4402   _E cep 4743   X. cxp 4832   dom cdm 4834   ran crn 4835   Oncon0 5423   -->wf 5578   -onto->wfo 5580   ` cfv 5582   1stc1st 6791   2ndc2nd 6792   Smo wsmo 7064  OrdIsocoi 8024  harchar 8071   ~<_^* cwdom 8072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-smo 7065  df-recs 7090  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-oi 8025  df-har 8073  df-wdom 8074

This theorem is referenced by:  hsmexlem3  8858