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Theorem iordsmo 6062
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 5131 . . 3  |-  (  _I  |`  A )  Fn  A
2 rnresi 4789 . . . 4  |-  ran  (  _I  |`  A )  =  A
3 iordsmo.1 . . . . 5  |-  Ord  A
4 ordsson 4309 . . . . 5  |-  ( Ord 
A  ->  A  C_  On )
53, 4ax-mp 7 . . . 4  |-  A  C_  On
62, 5eqsstri 3056 . . 3  |-  ran  (  _I  |`  A )  C_  On
7 df-f 5019 . . 3  |-  ( (  _I  |`  A ) : A --> On  <->  ( (  _I  |`  A )  Fn  A  /\  ran  (  _I  |`  A )  C_  On ) )
81, 6, 7mpbir2an 888 . 2  |-  (  _I  |`  A ) : A --> On
9 fvresi 5490 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
109adantr 270 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
11 fvresi 5490 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
1211adantl 271 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 y )  =  y )
1310, 12eleq12d 2158 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y )  <->  x  e.  y ) )
1413biimprd 156 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  e.  y  ->  ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y ) ) )
15 dmresi 4767 . 2  |-  dom  (  _I  |`  A )  =  A
168, 3, 14, 15issmo 6053 1  |-  Smo  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2999    _I cid 4115   Ord word 4189   Oncon0 4190   ran crn 4439    |` cres 4440    Fn wfn 5010   -->wf 5011   ` cfv 5015   Smo wsmo 6050
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-smo 6051
This theorem is referenced by:  smo0  6063
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