Theorem hsmexlem1 9248

Description: Lemma for hsmex 9254. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypothesis

Ref	Expression
hsmexlem.o	|- O = OrdIso ( _E , A )

Assertion

Ref	Expression
hsmexlem1	|- ( ( A C_ On /\ A ~<_* B ) -> dom O e. (har ` ~P B ) )

Proof of Theorem hsmexlem1

Step	Hyp	Ref	Expression
1		hsmexlem.o	. . . 4 |- O = OrdIso ( _E , A )
2	1	oicl 8434	. . 3 |- Ord dom O
3		relwdom 8471	. . . . . . . 8 |- Rel ~<_*
4	3	brrelexi 5158	. . . . . . 7 |- ( A ~<_* B -> A e. _V )
5	4	adantl 482	. . . . . 6 |- ( ( A C_ On /\ A ~<_* B ) -> A e. _V )
6		uniexg 6955	. . . . . 6 |- ( A e. _V -> U. A e. _V )
7		sucexg 7010	. . . . . 6 |- ( U. A e. _V -> suc U. A e. _V )
8	5, 6, 7	3syl 18	. . . . 5 |- ( ( A C_ On /\ A ~<_* B ) -> suc U. A e. _V )
9	1	oif 8435	. . . . . . 7 |- O : dom O --> A
10		onsucuni 7028	. . . . . . . 8 |- ( A C_ On -> A C_ suc U. A )
11	10	adantr 481	. . . . . . 7 |- ( ( A C_ On /\ A ~<_* B ) -> A C_ suc U. A )
12		fss 6056	. . . . . . 7 |- ( ( O : dom O --> A /\ A C_ suc U. A ) -> O : dom O --> suc U. A )
13	9, 11, 12	sylancr 695	. . . . . 6 |- ( ( A C_ On /\ A ~<_* B ) -> O : dom O --> suc U. A )
14	1	oismo 8445	. . . . . . . 8 |- ( A C_ On -> ( Smo O /\ ran O = A ) )
15	14	adantr 481	. . . . . . 7 |- ( ( A C_ On /\ A ~<_* B ) -> ( Smo O /\ ran O = A ) )
16	15	simpld 475	. . . . . 6 |- ( ( A C_ On /\ A ~<_* B ) -> Smo O )
17		ssorduni 6985	. . . . . . . 8 |- ( A C_ On -> Ord U. A )
18	17	adantr 481	. . . . . . 7 |- ( ( A C_ On /\ A ~<_* B ) -> Ord U. A )
19		ordsuc 7014	. . . . . . 7 |- ( Ord U. A <-> Ord suc U. A )
20	18, 19	sylib 208	. . . . . 6 |- ( ( A C_ On /\ A ~<_* B ) -> Ord suc U. A )
21		smorndom 7465	. . . . . 6 |- ( ( O : dom O --> suc U. A /\ Smo O /\ Ord suc U. A ) -> dom O C_ suc U. A )
22	13, 16, 20, 21	syl3anc 1326	. . . . 5 |- ( ( A C_ On /\ A ~<_* B ) -> dom O C_ suc U. A )
23	8, 22	ssexd 4805	. . . 4 |- ( ( A C_ On /\ A ~<_* B ) -> dom O e. _V )
24		elong 5731	. . . 4 |- ( dom O e. _V -> ( dom O e. On <-> Ord dom O ) )
25	23, 24	syl 17	. . 3 |- ( ( A C_ On /\ A ~<_* B ) -> ( dom O e. On <-> Ord dom O ) )
26	2, 25	mpbiri 248	. 2 |- ( ( A C_ On /\ A ~<_* B ) -> dom O e. On )
27		canth2g 8114	. . . 4 |- ( dom O e. _V -> dom O ~< ~P dom O )
28		sdomdom 7983	. . . 4 |- ( dom O ~< ~P dom O -> dom O ~<_ ~P dom O )
29	23, 27, 28	3syl 18	. . 3 |- ( ( A C_ On /\ A ~<_* B ) -> dom O ~<_ ~P dom O )
30		simpl 473	. . . . . . . . . . 11 |- ( ( A C_ On /\ A ~<_* B ) -> A C_ On )
31		epweon 6983	. . . . . . . . . . 11 |- _E We On
32		wess 5101	. . . . . . . . . . 11 |- ( A C_ On -> ( _E We On -> _E We A ) )
33	30, 31, 32	mpisyl 21	. . . . . . . . . 10 |- ( ( A C_ On /\ A ~<_* B ) -> _E We A )
34		epse 5097	. . . . . . . . . 10 |- _E Se A
35	1	oiiso2 8436	. . . . . . . . . 10 |- ( ( _E We A /\ _E Se A ) -> O Isom _E , _E ( dom O , ran O ) )
36	33, 34, 35	sylancl 694	. . . . . . . . 9 |- ( ( A C_ On /\ A ~<_* B ) -> O Isom _E , _E ( dom O , ran O ) )
37		isof1o 6573	. . . . . . . . 9 |- ( O Isom _E , _E ( dom O , ran O ) -> O : dom O -1-1-onto-> ran O )
38	36, 37	syl 17	. . . . . . . 8 |- ( ( A C_ On /\ A ~<_* B ) -> O : dom O -1-1-onto-> ran O )
39	15	simprd 479	. . . . . . . . 9 |- ( ( A C_ On /\ A ~<_* B ) -> ran O = A )
40		f1oeq3 6129	. . . . . . . . 9 |- ( ran O = A -> ( O : dom O -1-1-onto-> ran O <-> O : dom O -1-1-onto-> A ) )
41	39, 40	syl 17	. . . . . . . 8 |- ( ( A C_ On /\ A ~<_* B ) -> ( O : dom O -1-1-onto-> ran O <-> O : dom O -1-1-onto-> A ) )
42	38, 41	mpbid 222	. . . . . . 7 |- ( ( A C_ On /\ A ~<_* B ) -> O : dom O -1-1-onto-> A )
43		f1oen2g 7972	. . . . . . 7 |- ( ( dom O e. On /\ A e. _V /\ O : dom O -1-1-onto-> A ) -> dom O ~~ A )
44	26, 5, 42, 43	syl3anc 1326	. . . . . 6 |- ( ( A C_ On /\ A ~<_* B ) -> dom O ~~ A )
45		endom 7982	. . . . . 6 |- ( dom O ~~ A -> dom O ~<_ A )
46		domwdom 8479	. . . . . 6 |- ( dom O ~<_ A -> dom O ~<_* A )
47	44, 45, 46	3syl 18	. . . . 5 |- ( ( A C_ On /\ A ~<_* B ) -> dom O ~<_* A )
48		wdomtr 8480	. . . . 5 |- ( ( dom O ~<_* A /\ A ~<_* B ) -> dom O ~<_* B )
49	47, 48	sylancom 701	. . . 4 |- ( ( A C_ On /\ A ~<_* B ) -> dom O ~<_* B )
50		wdompwdom 8483	. . . 4 |- ( dom O ~<_* B -> ~P dom O ~<_ ~P B )
51	49, 50	syl 17	. . 3 |- ( ( A C_ On /\ A ~<_* B ) -> ~P dom O ~<_ ~P B )
52		domtr 8009	. . 3 |- ( ( dom O ~<_ ~P dom O /\ ~P dom O ~<_ ~P B ) -> dom O ~<_ ~P B )
53	29, 51, 52	syl2anc 693	. 2 |- ( ( A C_ On /\ A ~<_* B ) -> dom O ~<_ ~P B )
54		elharval 8468	. 2 |- ( dom O e. (har ` ~P B ) <-> ( dom O e. On /\ dom O ~<_ ~P B ) )
55	26, 53, 54	sylanbrc 698	1 |- ( ( A C_ On /\ A ~<_* B ) -> dom O e. (har ` ~P B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   _E cep 5028   Se wse 5071   We wwe 5072   dom cdm 5114   ran crn 5115   Ord word 5722   Oncon0 5723   suc csuc 5725   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   Smo wsmo 7442   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954  OrdIsocoi 8414  harchar 8461   ~<_^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-smo 7443  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-wdom 8464

This theorem is referenced by:  hsmexlem2  9249  hsmexlem4  9251