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Theorem iordsmo 6556
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 5503 . . 3  |-  (  _I  |`  A )  Fn  A
2 rnresi 5160 . . . 4  |-  ran  (  _I  |`  A )  =  A
3 iordsmo.1 . . . . 5  |-  Ord  A
4 ordsson 4711 . . . . 5  |-  ( Ord 
A  ->  A  C_  On )
53, 4ax-mp 8 . . . 4  |-  A  C_  On
62, 5eqsstri 3322 . . 3  |-  ran  (  _I  |`  A )  C_  On
7 df-f 5399 . . 3  |-  ( (  _I  |`  A ) : A --> On  <->  ( (  _I  |`  A )  Fn  A  /\  ran  (  _I  |`  A )  C_  On ) )
81, 6, 7mpbir2an 887 . 2  |-  (  _I  |`  A ) : A --> On
9 fvresi 5864 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
109adantr 452 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 x )  =  x )
11 fvresi 5864 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
1211adantl 453 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( (  _I  |`  A ) `
 y )  =  y )
1310, 12eleq12d 2456 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y )  <->  x  e.  y ) )
1413biimprd 215 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  e.  y  ->  ( (  _I  |`  A ) `  x
)  e.  ( (  _I  |`  A ) `  y ) ) )
15 dmresi 5137 . 2  |-  dom  (  _I  |`  A )  =  A
168, 3, 14, 15issmo 6547 1  |-  Smo  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264    _I cid 4435   Ord word 4522   Oncon0 4523   ran crn 4820    |` cres 4821    Fn wfn 5390   -->wf 5391   ` cfv 5395   Smo wsmo 6544
This theorem is referenced by:  smo0  6557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-smo 6545
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