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Theorem iordsmo 7341
 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 5922 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 rnresi 5398 . . . 4 ran ( I ↾ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 6881 . . . . 5 (Ord 𝐴𝐴 ⊆ On)
53, 4ax-mp 5 . . . 4 𝐴 ⊆ On
62, 5eqsstri 3598 . . 3 ran ( I ↾ 𝐴) ⊆ On
7 df-f 5808 . . 3 (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
81, 6, 7mpbir2an 957 . 2 ( I ↾ 𝐴):𝐴⟶On
9 fvresi 6344 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
109adantr 480 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11 fvresi 6344 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1211adantl 481 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1310, 12eleq12d 2682 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑦))
1413biimprd 237 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15 dmresi 5376 . 2 dom ( I ↾ 𝐴) = 𝐴
168, 3, 14, 15issmo 7332 1 Smo ( I ↾ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   I cid 4948  ran crn 5039   ↾ cres 5040  Ord word 5639  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Smo wsmo 7329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-smo 7330 This theorem is referenced by:  smo0  7342
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