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Theorem iordsmo 7988
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 6472 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 rnresi 5940 . . . 4 ran ( I ↾ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 7495 . . . . 5 (Ord 𝐴𝐴 ⊆ On)
53, 4ax-mp 5 . . . 4 𝐴 ⊆ On
62, 5eqsstri 4004 . . 3 ran ( I ↾ 𝐴) ⊆ On
7 df-f 6355 . . 3 (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
81, 6, 7mpbir2an 707 . 2 ( I ↾ 𝐴):𝐴⟶On
9 fvresi 6930 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
109adantr 481 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11 fvresi 6930 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1211adantl 482 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
1310, 12eleq12d 2911 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑦))
1413biimprd 249 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15 dmresi 5919 . 2 dom ( I ↾ 𝐴) = 𝐴
168, 3, 14, 15issmo 7979 1 Smo ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1530  wcel 2107  wss 3939   I cid 5457  ran crn 5554  cres 5555  Ord word 6187  Oncon0 6188   Fn wfn 6346  wf 6347  cfv 6351  Smo wsmo 7976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-smo 7977
This theorem is referenced by:  smo0  7989
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