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Theorem iordsmo 7318
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 5907 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 rnresi 5384 . . . 4 ran ( I ↾ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 6858 . . . . 5 (Ord 𝐴𝐴 ⊆ On)
53, 4ax-mp 5 . . . 4 𝐴 ⊆ On
62, 5eqsstri 3597 . . 3 ran ( I ↾ 𝐴) ⊆ On
7 df-f 5793 . . 3 (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
81, 6, 7mpbir2an 956 . 2 ( I ↾ 𝐴):𝐴⟶On
9 fvresi 6321 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
109adantr 479 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11 fvresi 6321 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1211adantl 480 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1310, 12eleq12d 2681 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑦))
1413biimprd 236 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15 dmresi 5362 . 2 dom ( I ↾ 𝐴) = 𝐴
168, 3, 14, 15issmo 7309 1 Smo ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1976  wss 3539   I cid 4937  ran crn 5028  cres 5029  Ord word 5624  Oncon0 5625   Fn wfn 5784  wf 5785  cfv 5789  Smo wsmo 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-ord 5628  df-on 5629  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-fv 5797  df-smo 7307
This theorem is referenced by:  smo0  7319
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