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 3988	. . . . . 6 |- ( B e. ~P On -> B C_ On )
2	1	adantr 466	. . . . 5 |- ( ( B e. ~P On /\ dom F e. C ) -> B C_ On )
3	2	ralimi 2818	. . . 4 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> A. a e. A B C_ On )
4		iunss 4337	. . . 4 |- ( U_ a e. A B C_ On <-> A. a e. A B C_ On )
5	3, 4	sylibr 215	. . 3 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> U_ a e. A B C_ On )
6	5	3ad2ant3 1028	. 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 6603	. . . 4 |- ( ( A e. _V /\ C e. On ) -> ( A X. C ) e. _V )
8	7	3adant3 1025	. . 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 1751	. . . . . . . . 9 |- F/ a C e. On
10		nfra1 2806	. . . . . . . . 9 |- F/ a A. a e. A ( B e. ~P On /\ dom F e. C )
11	9, 10	nfan 1984	. . . . . . . 8 |- F/ a ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) )
12		rsp 2791	. . . . . . . . 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 5480	. . . . . . . . . . . . . 14 |- ( C e. On -> ( dom F e. C -> dom F C_ C ) )
14	13	imp 430	. . . . . . . . . . . . 13 |- ( ( C e. On /\ dom F e. C ) -> dom F C_ C )
15	14	adantrl 720	. . . . . . . . . . . 12 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> dom F C_ C )
16	15	3adant3 1025	. . . . . . . . . . 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 8057	. . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . 16 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( Smo F /\ ran F = B ) )
21	20	simprd 464	. . . . . . . . . . . . . . 15 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ran F = B )
22	17	oif 8047	. . . . . . . . . . . . . . 15 |- F : dom F --> B
23	21, 22	jctil 539	. . . . . . . . . . . . . 14 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( F : dom F --> B /\ ran F = B ) )
24		dffo2 5810	. . . . . . . . . . . . . 14 |- ( F : dom F -onto-> B <-> ( F : dom F --> B /\ ran F = B ) )
25	23, 24	sylibr 215	. . . . . . . . . . . . 13 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> F : dom F -onto-> B )
26		dffo3 6048	. . . . . . . . . . . . . 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 465	. . . . . . . . . . . . 13 |- ( F : dom F -onto-> B -> A. b e. B E. e e. dom F b = ( F ` e ) )
28		rsp 2791	. . . . . . . . . . . . 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 1202	. . . . . . . . . . 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 3526	. . . . . . . . . . 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 1204	. . . . . . . . 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 662	. . . . . . . 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 2894	. . . . . . 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 1023	. . . . . 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 1751	. . . . . . 7 |- F/ d E. e e. C b = ( F ` e )
38		nfcv 2584	. . . . . . . 8 |- F/_ a C
39		nfcv 2584	. . . . . . . . . . 11 |- F/_ a _E
40		nfcsb1v 3411	. . . . . . . . . . 11 |- F/_ a [_ d / a ]_ B
41	39, 40	nfoi 8031	. . . . . . . . . 10 |- F/_ aOrdIso ( _E , [_ d / a ]_ B )
42		nfcv 2584	. . . . . . . . . 10 |- F/_ a e
43	41, 42	nffv 5884	. . . . . . . . 9 |- F/_ a (OrdIso ( _E , [_ d / a ]_ B ) ` e )
44	43	nfeq2 2601	. . . . . . . 8 |- F/ a b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
45	38, 44	nfrex 2888	. . . . . . 7 |- F/ a E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
46		csbeq1a 3404	. . . . . . . . . . . 12 |- ( a = d -> B = [_ d / a ]_ B )
47		oieq2 8030	. . . . . . . . . . . 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 2475	. . . . . . . . . 10 |- ( a = d -> F = OrdIso ( _E , [_ d / a ]_ B ) )
50	49	fveq1d 5879	. . . . . . . . 9 |- ( a = d -> ( F ` e ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
51	50	eqeq2d 2436	. . . . . . . 8 |- ( a = d -> ( b = ( F ` e ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
52	51	rexbidv 2939	. . . . . . 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 3052	. . . . . 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 229	. . . . 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 4301	. . . . 5 |- ( b e. U_ a e. A B <-> E. a e. A b e. B )
56		vex 3084	. . . . . . . . . . 11 |- d e. _V
57		vex 3084	. . . . . . . . . . 11 |- e e. _V
58	56, 57	op1std 6813	. . . . . . . . . 10 |- ( c = <. d , e >. -> ( 1st ` c ) = d )
59	58	csbeq1d 3402	. . . . . . . . 9 |- ( c = <. d , e >. -> [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B )
60		oieq2 8030	. . . . . . . . 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 6814	. . . . . . . 8 |- ( c = <. d , e >. -> ( 2nd ` c ) = e )
63	61, 62	fveq12d 5883	. . . . . . 7 |- ( c = <. d , e >. -> (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
64	63	eqeq2d 2436	. . . . . 6 |- ( c = <. d , e >. -> ( b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
65	64	rexxp 4992	. . . . 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 273	. . . 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 430	. . 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 8098	. 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 665	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 370   /\ w3a 982   = wceq 1437   e. wcel 1868   A.wral 2775   E.wrex 2776   _Vcvv 3081   [_csb 3395   C_ wss 3436   ~Pcpw 3979   <.cop 4002   U_ciun 4296   class class class wbr 4420   _E cep 4758   X. cxp 4847   dom cdm 4849   ran crn 4850   Oncon0 5438   -->wf 5593   -onto->wfo 5595   ` cfv 5597   1stc1st 6801   2ndc2nd 6802   Smo wsmo 7068  OrdIsocoi 8026  harchar 8073   ~<_^* cwdom 8074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-smo 7069  df-recs 7094  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-oi 8027  df-har 8075  df-wdom 8076

This theorem is referenced by:  hsmexlem3  8858