Theorem iordsmo 6385

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 5395	. . 3 |- ( _I |` A ) Fn A
2		rnresi 5040	. . . 4 |- ran ( _I |` A ) = A
3		iordsmo.1	. . . . 5 |- Ord A
4		ordsson 4541	. . . . 5 |- ( Ord A -> A C_ On )
5	3, 4	ax-mp 5	. . . 4 |- A C_ On
6	2, 5	eqsstri 3225	. . 3 |- ran ( _I |` A ) C_ On
7		df-f 5276	. . 3 |- ( ( _I |` A ) : A --> On <-> ( ( _I |` A ) Fn A /\ ran ( _I |` A ) C_ On ) )
8	1, 6, 7	mpbir2an 945	. 2 |- ( _I |` A ) : A --> On
9		fvresi 5779	. . . . 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 5779	. . . . 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 2276	. . 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 5015	. 2 |- dom ( _I |` A ) = A
16	8, 3, 14, 15	issmo 6376	1 |- Smo ( _I |` A )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   C_ wss 3166   _I cid 4336   Ord word 4410   Oncon0 4411   ran crn 4677   |` cres 4678   Fn wfn 5267   -->wf 5268   ` cfv 5272   Smo wsmo 6373

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-smo 6374

This theorem is referenced by:  smo0  6386