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Theorem iordsmo 6374
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 5361 . . 3  |-  (  _I  |`  A )  Fn  A
2 rnresi 5028 . . . 4  |-  ran  (  _I  |`  A )  =  A
3 iordsmo.1 . . . . 5  |-  Ord  A
4 ordsson 4581 . . . . 5  |-  ( Ord 
A  ->  A  C_  On )
53, 4ax-mp 8 . . . 4  |-  A  C_  On
62, 5eqsstri 3208 . . 3  |-  ran  (  _I  |`  A )  C_  On
7 df-f 5259 . . 3  |-  ( (  _I  |`  A ) : A --> On  <->  ( (  _I  |`  A )  Fn  A  /\  ran  (  _I  |`  A )  C_  On ) )
81, 6, 7mpbir2an 886 . 2  |-  (  _I  |`  A ) : A --> On
9 fvresi 5711 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
109adantr 451 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
11 fvresi 5711 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
1211adantl 452 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 y )  =  y )
1310, 12eleq12d 2351 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y )  <->  x  e.  y ) )
1413biimprd 214 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  e.  y  ->  ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y ) ) )
15 dmresi 5005 . 2  |-  dom  (  _I  |`  A )  =  A
168, 3, 14, 15issmo 6365 1  |-  Smo  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152    _I cid 4304   Ord word 4391   Oncon0 4392   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255   Smo wsmo 6362
This theorem is referenced by:  smo0  6375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-smo 6363
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