Theorem iordsmo 5912

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 5016	. . 3 |- ( _I |` A ) Fn A
2		rnresi 4682	. . . 4 |- ran ( _I |` A ) = A
3		iordsmo.1	. . . . 5 |- Ord A
4		ordsson 4218	. . . . 5 |- ( Ord A -> A C_ On )
5	3, 4	ax-mp 7	. . . 4 |- A C_ On
6	2, 5	eqsstri 2975	. . 3 |- ran ( _I |` A ) C_ On
7		df-f 4906	. . 3 |- ( ( _I |` A ) : A --> On <-> ( ( _I |` A ) Fn A /\ ran ( _I |` A ) C_ On ) )
8	1, 6, 7	mpbir2an 849	. 2 |- ( _I |` A ) : A --> On
9		fvresi 5356	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
10	9	adantr 261	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` x ) = x )
11		fvresi 5356	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
12	11	adantl 262	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` y ) = y )
13	10, 12	eleq12d 2108	. . 3 |- ( ( x e. A /\ y e. A ) -> ( ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) <-> x e. y ) )
14	13	biimprd 147	. 2 |- ( ( x e. A /\ y e. A ) -> ( x e. y -> ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) ) )
15		dmresi 4661	. 2 |- dom ( _I |` A ) = A
16	8, 3, 14, 15	issmo 5903	1 |- Smo ( _I |` A )

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   C_ wss 2917   _I cid 4025   Ord word 4099   Oncon0 4100   ran crn 4346   |` cres 4347   Fn wfn 4897   -->wf 4898   ` cfv 4902   Smo wsmo 5900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901

This theorem is referenced by:  smo0  5913