Theorem iordsmo 5935

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 5036	. . 3 |- ( _I |` A ) Fn A
2		rnresi 4702	. . . 4 |- ran ( _I |` A ) = A
3		iordsmo.1	. . . . 5 |- Ord A
4		ordsson 4236	. . . . 5 |- ( Ord A -> A C_ On )
5	3, 4	ax-mp 7	. . . 4 |- A C_ On
6	2, 5	eqsstri 3029	. . 3 |- ran ( _I |` A ) C_ On
7		df-f 4926	. . 3 |- ( ( _I |` A ) : A --> On <-> ( ( _I |` A ) Fn A /\ ran ( _I |` A ) C_ On ) )
8	1, 6, 7	mpbir2an 883	. 2 |- ( _I |` A ) : A --> On
9		fvresi 5377	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
10	9	adantr 270	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` x ) = x )
11		fvresi 5377	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
12	11	adantl 271	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` y ) = y )
13	10, 12	eleq12d 2149	. . 3 |- ( ( x e. A /\ y e. A ) -> ( ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) <-> x e. y ) )
14	13	biimprd 156	. 2 |- ( ( x e. A /\ y e. A ) -> ( x e. y -> ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) ) )
15		dmresi 4681	. 2 |- dom ( _I |` A ) = A
16	8, 3, 14, 15	issmo 5926	1 |- Smo ( _I |` A )

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   C_ wss 2973   _I cid 4043   Ord word 4117   Oncon0 4118   ran crn 4364   |` cres 4365   Fn wfn 4917   -->wf 4918   ` cfv 4922   Smo wsmo 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-smo 5924

This theorem is referenced by:  smo0  5936