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Theorem iordsmo 8304
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 6631 . . 3 ( I β†Ύ 𝐴) Fn 𝐴
2 rnresi 6028 . . . 4 ran ( I β†Ύ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 7718 . . . . 5 (Ord 𝐴 β†’ 𝐴 βŠ† On)
53, 4ax-mp 5 . . . 4 𝐴 βŠ† On
62, 5eqsstri 3979 . . 3 ran ( I β†Ύ 𝐴) βŠ† On
7 df-f 6501 . . 3 (( I β†Ύ 𝐴):𝐴⟢On ↔ (( I β†Ύ 𝐴) Fn 𝐴 ∧ ran ( I β†Ύ 𝐴) βŠ† On))
81, 6, 7mpbir2an 710 . 2 ( I β†Ύ 𝐴):𝐴⟢On
9 fvresi 7120 . . . . 5 (π‘₯ ∈ 𝐴 β†’ (( I β†Ύ 𝐴)β€˜π‘₯) = π‘₯)
109adantr 482 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (( I β†Ύ 𝐴)β€˜π‘₯) = π‘₯)
11 fvresi 7120 . . . . 5 (𝑦 ∈ 𝐴 β†’ (( I β†Ύ 𝐴)β€˜π‘¦) = 𝑦)
1211adantl 483 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (( I β†Ύ 𝐴)β€˜π‘¦) = 𝑦)
1310, 12eleq12d 2828 . . 3 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ ((( I β†Ύ 𝐴)β€˜π‘₯) ∈ (( I β†Ύ 𝐴)β€˜π‘¦) ↔ π‘₯ ∈ 𝑦))
1413biimprd 248 . 2 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ ∈ 𝑦 β†’ (( I β†Ύ 𝐴)β€˜π‘₯) ∈ (( I β†Ύ 𝐴)β€˜π‘¦)))
15 dmresi 6006 . 2 dom ( I β†Ύ 𝐴) = 𝐴
168, 3, 14, 15issmo 8295 1 Smo ( I β†Ύ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911   I cid 5531  ran crn 5635   β†Ύ cres 5636  Ord word 6317  Oncon0 6318   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  Smo wsmo 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-smo 8293
This theorem is referenced by:  smo0  8305
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