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Theorem iordsmo 6985
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 5635 . . 3  |-  (  _I  |`  A )  Fn  A
2 rnresi 5291 . . . 4  |-  ran  (  _I  |`  A )  =  A
3 iordsmo.1 . . . . 5  |-  Ord  A
4 ordsson 6563 . . . . 5  |-  ( Ord 
A  ->  A  C_  On )
53, 4ax-mp 5 . . . 4  |-  A  C_  On
62, 5eqsstri 3471 . . 3  |-  ran  (  _I  |`  A )  C_  On
7 df-f 5529 . . 3  |-  ( (  _I  |`  A ) : A --> On  <->  ( (  _I  |`  A )  Fn  A  /\  ran  (  _I  |`  A )  C_  On ) )
81, 6, 7mpbir2an 921 . 2  |-  (  _I  |`  A ) : A --> On
9 fvresi 6033 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
109adantr 463 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
11 fvresi 6033 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
1211adantl 464 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 y )  =  y )
1310, 12eleq12d 2484 . . 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 5270 . 2  |-  dom  (  _I  |`  A )  =  A
168, 3, 14, 15issmo 6976 1  |-  Smo  (  _I  |`  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3413    _I cid 4732   Ord word 4820   Oncon0 4821   ran crn 4943    |` cres 4944    Fn wfn 5520   -->wf 5521   ` cfv 5525   Smo wsmo 6973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-smo 6974
This theorem is referenced by:  smo0  6986
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