Theorem oiid 8446

Description: The order type of an ordinal under the e. order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
oiid	|- ( Ord A -> OrdIso ( _E , A ) = ( _I |` A ) )

Proof of Theorem oiid

Step	Hyp	Ref	Expression
1		ordwe 5736	. 2 |- ( Ord A -> _E We A )
2		epse 5097	. . 3 |- _E Se A
3	2	a1i 11	. 2 |- ( Ord A -> _E Se A )
4		eqid 2622	. . . . . 6 |- OrdIso ( _E , A ) = OrdIso ( _E , A )
5	4	oiiso2 8436	. . . . 5 |- ( ( _E We A /\ _E Se A ) -> OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , ran OrdIso ( _E , A ) ) )
6	1, 2, 5	sylancl 694	. . . 4 |- ( Ord A -> OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , ran OrdIso ( _E , A ) ) )
7		ordsson 6989	. . . . . . 7 |- ( Ord A -> A C_ On )
8	4	oismo 8445	. . . . . . 7 |- ( A C_ On -> ( Smo OrdIso ( _E , A ) /\ ran OrdIso ( _E , A ) = A ) )
9	7, 8	syl 17	. . . . . 6 |- ( Ord A -> ( Smo OrdIso ( _E , A ) /\ ran OrdIso ( _E , A ) = A ) )
10	9	simprd 479	. . . . 5 |- ( Ord A -> ran OrdIso ( _E , A ) = A )
11		isoeq5 6571	. . . . 5 |- ( ran OrdIso ( _E , A ) = A -> (OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , ran OrdIso ( _E , A ) ) <-> OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , A ) ) )
12	10, 11	syl 17	. . . 4 |- ( Ord A -> (OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , ran OrdIso ( _E , A ) ) <-> OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , A ) ) )
13	6, 12	mpbid 222	. . 3 |- ( Ord A -> OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , A ) )
14	4	oicl 8434	. . . . . 6 |- Ord dom OrdIso ( _E , A )
15	14	a1i 11	. . . . 5 |- ( Ord A -> Ord dom OrdIso ( _E , A ) )
16		id 22	. . . . 5 |- ( Ord A -> Ord A )
17		ordiso2 8420	. . . . 5 |- ( (OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , A ) /\ Ord dom OrdIso ( _E , A ) /\ Ord A ) -> dom OrdIso ( _E , A ) = A )
18	13, 15, 16, 17	syl3anc 1326	. . . 4 |- ( Ord A -> dom OrdIso ( _E , A ) = A )
19		isoeq4 6570	. . . 4 |- ( dom OrdIso ( _E , A ) = A -> (OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , A ) <-> OrdIso ( _E , A ) Isom _E , _E ( A , A ) ) )
20	18, 19	syl 17	. . 3 |- ( Ord A -> (OrdIso ( _E , A ) Isom _E , _E ( dom OrdIso ( _E , A ) , A ) <-> OrdIso ( _E , A ) Isom _E , _E ( A , A ) ) )
21	13, 20	mpbid 222	. 2 |- ( Ord A -> OrdIso ( _E , A ) Isom _E , _E ( A , A ) )
22		weniso 6604	. 2 |- ( ( _E We A /\ _E Se A /\ OrdIso ( _E , A ) Isom _E , _E ( A , A ) ) -> OrdIso ( _E , A ) = ( _I |` A ) )
23	1, 3, 21, 22	syl3anc 1326	1 |- ( Ord A -> OrdIso ( _E , A ) = ( _I |` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   _I cid 5023   _E cep 5028   Se wse 5071   We wwe 5072   dom cdm 5114   ran crn 5115   |` cres 5116   Ord word 5722   Oncon0 5723   Isom wiso 5889   Smo wsmo 7442  OrdIsocoi 8414

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-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-oi 8415

This theorem is referenced by:  hsmexlem5  9252