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Theorem iordsmo 5825
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 4933 . . 3 ( I ↾ A) Fn A
2 rnresi 4600 . . . 4 ran ( I ↾ A) = A
3 iordsmo.1 . . . . 5 Ord A
4 ordsson 4159 . . . . 5 (Ord AA ⊆ On)
53, 4ax-mp 7 . . . 4 A ⊆ On
62, 5eqsstri 2946 . . 3 ran ( I ↾ A) ⊆ On
7 df-f 4824 . . 3 (( I ↾ A):A⟶On ↔ (( I ↾ A) Fn A ran ( I ↾ A) ⊆ On))
81, 6, 7mpbir2an 833 . 2 ( I ↾ A):A⟶On
9 fvresi 5272 . . . . 5 (x A → (( I ↾ A)‘x) = x)
109adantr 261 . . . 4 ((x A y A) → (( I ↾ A)‘x) = x)
11 fvresi 5272 . . . . 5 (y A → (( I ↾ A)‘y) = y)
1211adantl 262 . . . 4 ((x A y A) → (( I ↾ A)‘y) = y)
1310, 12eleq12d 2084 . . 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 4579 . 2 dom ( I ↾ A) = A
168, 3, 14, 15issmo 5816 1 Smo ( I ↾ A)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226   wcel 1369  wss 2888   I cid 3991  Ord word 4040  Oncon0 4041  ran crn 4264  cres 4265   Fn wfn 4815  wf 4816  cfv 4820  Smo wsmo 5813
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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-14 1381  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-sep 3841  ax-pow 3893  ax-pr 3910
This theorem depends on definitions:  df-bi 110  df-3an 871  df-tru 1229  df-nf 1326  df-sb 1622  df-eu 1879  df-mo 1880  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ral 2283  df-rex 2284  df-v 2531  df-sbc 2736  df-un 2893  df-in 2895  df-ss 2902  df-pw 3328  df-sn 3348  df-pr 3349  df-op 3351  df-uni 3547  df-br 3731  df-opab 3785  df-tr 3821  df-id 3996  df-iord 4044  df-on 4046  df-xp 4269  df-rel 4270  df-cnv 4271  df-co 4272  df-dm 4273  df-rn 4274  df-res 4275  df-ima 4276  df-iota 4785  df-fun 4822  df-fn 4823  df-f 4824  df-fv 4828  df-smo 5814
This theorem is referenced by:  smo0  5826
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