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Theorem iordsmo 8384
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 β†Ύ 𝐴)

Proof of Theorem iordsmo
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnresi 6689 . . 3 ( I β†Ύ 𝐴) Fn 𝐴
2 rnresi 6083 . . . 4 ran ( I β†Ύ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 7791 . . . . 5 (Ord 𝐴 β†’ 𝐴 βŠ† On)
53, 4ax-mp 5 . . . 4 𝐴 βŠ† On
62, 5eqsstri 4016 . . 3 ran ( I β†Ύ 𝐴) βŠ† On
7 df-f 6557 . . 3 (( I β†Ύ 𝐴):𝐴⟢On ↔ (( I β†Ύ 𝐴) Fn 𝐴 ∧ ran ( I β†Ύ 𝐴) βŠ† On))
81, 6, 7mpbir2an 709 . 2 ( I β†Ύ 𝐴):𝐴⟢On
9 fvresi 7188 . . . . 5 (π‘₯ ∈ 𝐴 β†’ (( I β†Ύ 𝐴)β€˜π‘₯) = π‘₯)
109adantr 479 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (( I β†Ύ 𝐴)β€˜π‘₯) = π‘₯)
11 fvresi 7188 . . . . 5 (𝑦 ∈ 𝐴 β†’ (( I β†Ύ 𝐴)β€˜π‘¦) = 𝑦)
1211adantl 480 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (( I β†Ύ 𝐴)β€˜π‘¦) = 𝑦)
1310, 12eleq12d 2823 . . 3 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ ((( I β†Ύ 𝐴)β€˜π‘₯) ∈ (( I β†Ύ 𝐴)β€˜π‘¦) ↔ π‘₯ ∈ 𝑦))
1413biimprd 247 . 2 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ ∈ 𝑦 β†’ (( I β†Ύ 𝐴)β€˜π‘₯) ∈ (( I β†Ύ 𝐴)β€˜π‘¦)))
15 dmresi 6060 . 2 dom ( I β†Ύ 𝐴) = 𝐴
168, 3, 14, 15issmo 8375 1 Smo ( I β†Ύ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949   I cid 5579  ran crn 5683   β†Ύ cres 5684  Ord word 6373  Oncon0 6374   Fn wfn 6548  βŸΆwf 6549  β€˜cfv 6553  Smo wsmo 8372
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-smo 8373
This theorem is referenced by:  smo0  8385
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