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Theorem iordsmo 5853
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 4959 . . 3 ( I ↾ A) Fn A
2 rnresi 4625 . . . 4 ran ( I ↾ A) = A
3 iordsmo.1 . . . . 5 Ord A
4 ordsson 4184 . . . . 5 (Ord AA ⊆ On)
53, 4ax-mp 7 . . . 4 A ⊆ On
62, 5eqsstri 2969 . . 3 ran ( I ↾ A) ⊆ On
7 df-f 4849 . . 3 (( I ↾ A):A⟶On ↔ (( I ↾ A) Fn A ran ( I ↾ A) ⊆ On))
81, 6, 7mpbir2an 848 . 2 ( I ↾ A):A⟶On
9 fvresi 5299 . . . . 5 (x A → (( I ↾ A)‘x) = x)
109adantr 261 . . . 4 ((x A y A) → (( I ↾ A)‘x) = x)
11 fvresi 5299 . . . . 5 (y A → (( I ↾ A)‘y) = y)
1211adantl 262 . . . 4 ((x A y A) → (( I ↾ A)‘y) = y)
1310, 12eleq12d 2105 . . 3 ((x A y A) → ((( I ↾ A)‘x) (( I ↾ A)‘y) ↔ x y))
1413biimprd 147 . 2 ((x A y A) → (x y → (( I ↾ A)‘x) (( I ↾ A)‘y)))
15 dmresi 4604 . 2 dom ( I ↾ A) = A
168, 3, 14, 15issmo 5844 1 Smo ( I ↾ A)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  wss 2911   I cid 4016  Ord word 4065  Oncon0 4066  ran crn 4289  cres 4290   Fn wfn 4840  wf 4841  cfv 4845  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-smo 5842
This theorem is referenced by:  smo0  5854
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