Theorem hsmexlem2 8596

Description: Lemma for hsmex 8601. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8739 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 3869	. . . . . 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 2791	. . . 4 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> A. a e. A B C_ On )
4		iunss 4211	. . . 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 1011	. 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 6507	. . . 4 |- ( ( A e. _V /\ C e. On ) -> ( A X. C ) e. _V )
8	7	3adant3 1008	. . 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 1673	. . . . . . . . 9 |- F/ a C e. On
10		nfra1 2766	. . . . . . . . 9 |- F/ a A. a e. A ( B e. ~P On /\ dom F e. C )
11	9, 10	nfan 1861	. . . . . . . 8 |- F/ a ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) )
12		rsp 2776	. . . . . . . . 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 4761	. . . . . . . . . . . . . 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 1008	. . . . . . . . . . 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 7754	. . . . . . . . . . . . . . . . . 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 7744	. . . . . . . . . . . . . . 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 5624	. . . . . . . . . . . . . 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 5858	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 1184	. . . . . . . . . . 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 3417	. . . . . . . . . . 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 1186	. . . . . . . . 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 2824	. . . . . . 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 1006	. . . . . 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 1673	. . . . . . 7 |- F/ d E. e e. C b = ( F ` e )
38		nfcv 2579	. . . . . . . 8 |- F/_ a C
39		nfcv 2579	. . . . . . . . . . 11 |- F/_ a _E
40		nfcsb1v 3304	. . . . . . . . . . 11 |- F/_ a [_ d / a ]_ B
41	39, 40	nfoi 7728	. . . . . . . . . 10 |- F/_ aOrdIso ( _E , [_ d / a ]_ B )
42		nfcv 2579	. . . . . . . . . 10 |- F/_ a e
43	41, 42	nffv 5698	. . . . . . . . 9 |- F/_ a (OrdIso ( _E , [_ d / a ]_ B ) ` e )
44	43	nfeq2 2590	. . . . . . . 8 |- F/ a b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
45	38, 44	nfrex 2771	. . . . . . 7 |- F/ a E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
46		csbeq1a 3297	. . . . . . . . . . . 12 |- ( a = d -> B = [_ d / a ]_ B )
47		oieq2 7727	. . . . . . . . . . . 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 2487	. . . . . . . . . 10 |- ( a = d -> F = OrdIso ( _E , [_ d / a ]_ B ) )
50	49	fveq1d 5693	. . . . . . . . 9 |- ( a = d -> ( F ` e ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
51	50	eqeq2d 2454	. . . . . . . 8 |- ( a = d -> ( b = ( F ` e ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
52	51	rexbidv 2736	. . . . . . 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 2944	. . . . . 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 4175	. . . . 5 |- ( b e. U_ a e. A B <-> E. a e. A b e. B )
56		vex 2975	. . . . . . . . . . 11 |- d e. _V
57		vex 2975	. . . . . . . . . . 11 |- e e. _V
58	56, 57	op1std 6587	. . . . . . . . . 10 |- ( c = <. d , e >. -> ( 1st ` c ) = d )
59	58	csbeq1d 3295	. . . . . . . . 9 |- ( c = <. d , e >. -> [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B )
60		oieq2 7727	. . . . . . . . 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 6588	. . . . . . . 8 |- ( c = <. d , e >. -> ( 2nd ` c ) = e )
63	61, 62	fveq12d 5697	. . . . . . 7 |- ( c = <. d , e >. -> (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
64	63	eqeq2d 2454	. . . . . 6 |- ( c = <. d , e >. -> ( b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
65	64	rexxp 4982	. . . . 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 7796	. 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 8595	. 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 965   = wceq 1369   e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972   [_csb 3288   C_ wss 3328   ~Pcpw 3860   <.cop 3883   U_ciun 4171   class class class wbr 4292   _E cep 4630   Oncon0 4719   X. cxp 4838   dom cdm 4840   ran crn 4841   -->wf 5414   -onto->wfo 5416   ` cfv 5418   1stc1st 6575   2ndc2nd 6576   Smo wsmo 6806  OrdIsocoi 7723  harchar 7771   ~<_^* cwdom 7772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-1st 6577  df-2nd 6578  df-smo 6807  df-recs 6832  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-oi 7724  df-har 7773  df-wdom 7774

This theorem is referenced by:  hsmexlem3  8597