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Theorem iordsmo 5912
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 5016 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 rnresi 4682 . . . 4 ran ( I ↾ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 4218 . . . . 5 (Ord 𝐴𝐴 ⊆ On)
53, 4ax-mp 7 . . . 4 𝐴 ⊆ On
62, 5eqsstri 2975 . . 3 ran ( I ↾ 𝐴) ⊆ On
7 df-f 4906 . . 3 (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
81, 6, 7mpbir2an 849 . 2 ( I ↾ 𝐴):𝐴⟶On
9 fvresi 5356 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
109adantr 261 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11 fvresi 5356 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1211adantl 262 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1310, 12eleq12d 2108 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑦))
1413biimprd 147 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15 dmresi 4661 . 2 dom ( I ↾ 𝐴) = 𝐴
168, 3, 14, 15issmo 5903 1 Smo ( I ↾ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wcel 1393  wss 2917   I cid 4025  Ord word 4099  Oncon0 4100  ran crn 4346  cres 4347   Fn wfn 4897  wf 4898  cfv 4902  Smo wsmo 5900
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901
This theorem is referenced by:  smo0  5913
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