Theorem iordsmo 6316

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 5348	. . 3 |- ( _I |` A ) Fn A
2		rnresi 5000	. . . 4 |- ran ( _I |` A ) = A
3		iordsmo.1	. . . . 5 |- Ord A
4		ordsson 4506	. . . . 5 |- ( Ord A -> A C_ On )
5	3, 4	ax-mp 5	. . . 4 |- A C_ On
6	2, 5	eqsstri 3202	. . 3 |- ran ( _I |` A ) C_ On
7		df-f 5235	. . 3 |- ( ( _I |` A ) : A --> On <-> ( ( _I |` A ) Fn A /\ ran ( _I |` A ) C_ On ) )
8	1, 6, 7	mpbir2an 944	. 2 |- ( _I |` A ) : A --> On
9		fvresi 5725	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
10	9	adantr 276	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` x ) = x )
11		fvresi 5725	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
12	11	adantl 277	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( _I |` A ) ` y ) = y )
13	10, 12	eleq12d 2260	. . 3 |- ( ( x e. A /\ y e. A ) -> ( ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) <-> x e. y ) )
14	13	biimprd 158	. 2 |- ( ( x e. A /\ y e. A ) -> ( x e. y -> ( ( _I |` A ) ` x ) e. ( ( _I |` A ) ` y ) ) )
15		dmresi 4977	. 2 |- dom ( _I |` A ) = A
16	8, 3, 14, 15	issmo 6307	1 |- Smo ( _I |` A )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2160   C_ wss 3144   _I cid 4303   Ord word 4377   Oncon0 4378   ran crn 4642   |` cres 4643   Fn wfn 5226   -->wf 5227   ` cfv 5231   Smo wsmo 6304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-smo 6305

This theorem is referenced by:  smo0  6317