Theorem hsmexlem2 8798

Description: Lemma for hsmex 8803. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8941 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 4008	. . . . . 6 |- ( B e. ~P On -> B C_ On )
2	1	adantr 463	. . . . 5 |- ( ( B e. ~P On /\ dom F e. C ) -> B C_ On )
3	2	ralimi 2847	. . . 4 |- ( A. a e. A ( B e. ~P On /\ dom F e. C ) -> A. a e. A B C_ On )
4		iunss 4356	. . . 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 1017	. 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 6575	. . . 4 |- ( ( A e. _V /\ C e. On ) -> ( A X. C ) e. _V )
8	7	3adant3 1014	. . 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 1712	. . . . . . . . 9 |- F/ a C e. On
10		nfra1 2835	. . . . . . . . 9 |- F/ a A. a e. A ( B e. ~P On /\ dom F e. C )
11	9, 10	nfan 1933	. . . . . . . 8 |- F/ a ( C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) )
12		rsp 2820	. . . . . . . . 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 4909	. . . . . . . . . . . . . 14 |- ( C e. On -> ( dom F e. C -> dom F C_ C ) )
14	13	imp 427	. . . . . . . . . . . . 13 |- ( ( C e. On /\ dom F e. C ) -> dom F C_ C )
15	14	adantrl 713	. . . . . . . . . . . 12 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> dom F C_ C )
16	15	3adant3 1014	. . . . . . . . . . 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 7957	. . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . 16 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( Smo F /\ ran F = B ) )
21	20	simprd 461	. . . . . . . . . . . . . . 15 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ran F = B )
22	17	oif 7947	. . . . . . . . . . . . . . 15 |- F : dom F --> B
23	21, 22	jctil 535	. . . . . . . . . . . . . 14 |- ( ( C e. On /\ ( B e. ~P On /\ dom F e. C ) ) -> ( F : dom F --> B /\ ran F = B ) )
24		dffo2 5781	. . . . . . . . . . . . . 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 6022	. . . . . . . . . . . . . 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 462	. . . . . . . . . . . . 13 |- ( F : dom F -onto-> B -> A. b e. B E. e e. dom F b = ( F ` e ) )
28		rsp 2820	. . . . . . . . . . . . 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 1191	. . . . . . . . . . 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 3551	. . . . . . . . . . 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 1193	. . . . . . . . 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 656	. . . . . . . 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 2923	. . . . . . 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 1012	. . . . . 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 1712	. . . . . . 7 |- F/ d E. e e. C b = ( F ` e )
38		nfcv 2616	. . . . . . . 8 |- F/_ a C
39		nfcv 2616	. . . . . . . . . . 11 |- F/_ a _E
40		nfcsb1v 3436	. . . . . . . . . . 11 |- F/_ a [_ d / a ]_ B
41	39, 40	nfoi 7931	. . . . . . . . . 10 |- F/_ aOrdIso ( _E , [_ d / a ]_ B )
42		nfcv 2616	. . . . . . . . . 10 |- F/_ a e
43	41, 42	nffv 5855	. . . . . . . . 9 |- F/_ a (OrdIso ( _E , [_ d / a ]_ B ) ` e )
44	43	nfeq2 2633	. . . . . . . 8 |- F/ a b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
45	38, 44	nfrex 2917	. . . . . . 7 |- F/ a E. e e. C b = (OrdIso ( _E , [_ d / a ]_ B ) ` e )
46		csbeq1a 3429	. . . . . . . . . . . 12 |- ( a = d -> B = [_ d / a ]_ B )
47		oieq2 7930	. . . . . . . . . . . 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 2507	. . . . . . . . . 10 |- ( a = d -> F = OrdIso ( _E , [_ d / a ]_ B ) )
50	49	fveq1d 5850	. . . . . . . . 9 |- ( a = d -> ( F ` e ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
51	50	eqeq2d 2468	. . . . . . . 8 |- ( a = d -> ( b = ( F ` e ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
52	51	rexbidv 2965	. . . . . . 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 3078	. . . . . 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 4320	. . . . 5 |- ( b e. U_ a e. A B <-> E. a e. A b e. B )
56		vex 3109	. . . . . . . . . . 11 |- d e. _V
57		vex 3109	. . . . . . . . . . 11 |- e e. _V
58	56, 57	op1std 6783	. . . . . . . . . 10 |- ( c = <. d , e >. -> ( 1st ` c ) = d )
59	58	csbeq1d 3427	. . . . . . . . 9 |- ( c = <. d , e >. -> [_ ( 1st ` c ) / a ]_ B = [_ d / a ]_ B )
60		oieq2 7930	. . . . . . . . 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 6784	. . . . . . . 8 |- ( c = <. d , e >. -> ( 2nd ` c ) = e )
63	61, 62	fveq12d 5854	. . . . . . 7 |- ( c = <. d , e >. -> (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) )
64	63	eqeq2d 2468	. . . . . 6 |- ( c = <. d , e >. -> ( b = (OrdIso ( _E , [_ ( 1st ` c ) / a ]_ B ) ` ( 2nd ` c ) ) <-> b = (OrdIso ( _E , [_ d / a ]_ B ) ` e ) ) )
65	64	rexxp 5134	. . . . 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 427	. . 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 7999	. 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 8797	. 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 659	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106   [_csb 3420   C_ wss 3461   ~Pcpw 3999   <.cop 4022   U_ciun 4315   class class class wbr 4439   _E cep 4778   Oncon0 4867   X. cxp 4986   dom cdm 4988   ran crn 4989   -->wf 5566   -onto->wfo 5568   ` cfv 5570   1stc1st 6771   2ndc2nd 6772   Smo wsmo 7008  OrdIsocoi 7926  harchar 7974   ~<_^* cwdom 7975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-1st 6773  df-2nd 6774  df-smo 7009  df-recs 7034  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-oi 7927  df-har 7976  df-wdom 7977

This theorem is referenced by:  hsmexlem3  8799