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Theorem iordsmo 5834
 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 4942 . . 3 ( I ↾ A) Fn A
2 rnresi 4609 . . . 4 ran ( I ↾ A) = A
3 iordsmo.1 . . . . 5 Ord A
4 ordsson 4168 . . . . 5 (Ord AA ⊆ On)
53, 4ax-mp 7 . . . 4 A ⊆ On
62, 5eqsstri 2952 . . 3 ran ( I ↾ A) ⊆ On
7 df-f 4833 . . 3 (( I ↾ A):A⟶On ↔ (( I ↾ A) Fn A ran ( I ↾ A) ⊆ On))
81, 6, 7mpbir2an 837 . 2 ( I ↾ A):A⟶On
9 fvresi 5281 . . . . 5 (x A → (( I ↾ A)‘x) = x)
109adantr 261 . . . 4 ((x A y A) → (( I ↾ A)‘x) = x)
11 fvresi 5281 . . . . 5 (y A → (( I ↾ A)‘y) = y)
1211adantl 262 . . . 4 ((x A y A) → (( I ↾ A)‘y) = y)
1310, 12eleq12d 2090 . . 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 4588 . 2 dom ( I ↾ A) = A
168, 3, 14, 15issmo 5825 1 Smo ( I ↾ A)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894   I cid 3999  Ord word 4048  Oncon0 4049  ran crn 4273   ↾ cres 4274   Fn wfn 4824  ⟶wf 4825  ‘cfv 4829  Smo wsmo 5822 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fv 4837  df-smo 5823 This theorem is referenced by:  smo0  5835
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