MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iordsmo Structured version   Unicode version

Theorem iordsmo 7025
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.
StepHypRef Expression
1 fnresi 5696 . . 3  |-  (  _I  |`  A )  Fn  A
2 rnresi 5348 . . . 4  |-  ran  (  _I  |`  A )  =  A
3 iordsmo.1 . . . . 5  |-  Ord  A
4 ordsson 6603 . . . . 5  |-  ( Ord 
A  ->  A  C_  On )
53, 4ax-mp 5 . . . 4  |-  A  C_  On
62, 5eqsstri 3534 . . 3  |-  ran  (  _I  |`  A )  C_  On
7 df-f 5590 . . 3  |-  ( (  _I  |`  A ) : A --> On  <->  ( (  _I  |`  A )  Fn  A  /\  ran  (  _I  |`  A )  C_  On ) )
81, 6, 7mpbir2an 918 . 2  |-  (  _I  |`  A ) : A --> On
9 fvresi 6085 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
109adantr 465 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
11 fvresi 6085 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
1211adantl 466 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 y )  =  y )
1310, 12eleq12d 2549 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y )  <->  x  e.  y ) )
1413biimprd 223 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  e.  y  ->  ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y ) ) )
15 dmresi 5327 . 2  |-  dom  (  _I  |`  A )  =  A
168, 3, 14, 15issmo 7016 1  |-  Smo  (  _I  |`  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476    _I cid 4790   Ord word 4877   Oncon0 4878   ran crn 5000    |` cres 5001    Fn wfn 5581   -->wf 5582   ` cfv 5586   Smo wsmo 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-smo 7014
This theorem is referenced by:  smo0  7026
  Copyright terms: Public domain W3C validator