Theorem iordsmo 6202

Description: The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)

Hypothesis

Ref	Expression
iordsmo.1	|- Ord A

Assertion

Ref	Expression
iordsmo	|- Smo ( _I |` A )

Proof of Theorem iordsmo

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnresi 5248	. . 3 |- ( _I |` A ) Fn A
2		rnresi 4904	. . . 4 |- ran ( _I |` A ) = A
3		iordsmo.1	. . . . 5 |- Ord A
4		ordsson 4416	. . . . 5 |- ( Ord A -> A C_ On )
5	3, 4	ax-mp 5	. . . 4 |- A C_ On
6	2, 5	eqsstri 3134	. . 3 |- ran ( _I |` A ) C_ On
7		df-f 5135	. . 3 |- ( ( _I |` A ) : A --> On <-> ( ( _I |` A ) Fn A /\ ran ( _I |` A ) C_ On ) )
8	1, 6, 7	mpbir2an 927	. 2 |- ( _I |` A ) : A --> On
9		fvresi 5621	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
10	9	adantr 274	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` x ) = x )
11		fvresi 5621	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
12	11	adantl 275	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` y ) = y )
13	10, 12	eleq12d 2211	. . 3 |- ( ( x e. A /\ y e. A ) -> ( ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) <-> x e. y ) )
14	13	biimprd 157	. 2 |- ( ( x e. A /\ y e. A ) -> ( x e. y -> ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) ) )
15		dmresi 4882	. 2 |- dom ( _I |` A ) = A
16	8, 3, 14, 15	issmo 6193	1 |- Smo ( _I |` A )

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 1481   C_ wss 3076   _I cid 4218   Ord word 4292   Oncon0 4293   ran crn 4548   |` cres 4549   Fn wfn 5126   -->wf 5127   ` cfv 5131   Smo wsmo 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-smo 6191

This theorem is referenced by:  smo0  6203