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Theorem iordsmo 6124
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 5176 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 rnresi 4832 . . . 4 ran ( I ↾ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 4346 . . . . 5 (Ord 𝐴𝐴 ⊆ On)
53, 4ax-mp 7 . . . 4 𝐴 ⊆ On
62, 5eqsstri 3079 . . 3 ran ( I ↾ 𝐴) ⊆ On
7 df-f 5063 . . 3 (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
81, 6, 7mpbir2an 894 . 2 ( I ↾ 𝐴):𝐴⟶On
9 fvresi 5545 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
109adantr 272 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11 fvresi 5545 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1211adantl 273 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1310, 12eleq12d 2170 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑦))
1413biimprd 157 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15 dmresi 4810 . 2 dom ( I ↾ 𝐴) = 𝐴
168, 3, 14, 15issmo 6115 1 Smo ( I ↾ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1299  wcel 1448  wss 3021   I cid 4148  Ord word 4222  Oncon0 4223  ran crn 4478  cres 4479   Fn wfn 5054  wf 5055  cfv 5059  Smo wsmo 6112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-smo 6113
This theorem is referenced by:  smo0  6125
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