MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iordsmo Structured version   Visualization version   GIF version

Theorem iordsmo 8355
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 6672 . . 3 ( I β†Ύ 𝐴) Fn 𝐴
2 rnresi 6067 . . . 4 ran ( I β†Ύ 𝐴) = 𝐴
3 iordsmo.1 . . . . 5 Ord 𝐴
4 ordsson 7766 . . . . 5 (Ord 𝐴 β†’ 𝐴 βŠ† On)
53, 4ax-mp 5 . . . 4 𝐴 βŠ† On
62, 5eqsstri 4011 . . 3 ran ( I β†Ύ 𝐴) βŠ† On
7 df-f 6540 . . 3 (( I β†Ύ 𝐴):𝐴⟢On ↔ (( I β†Ύ 𝐴) Fn 𝐴 ∧ ran ( I β†Ύ 𝐴) βŠ† On))
81, 6, 7mpbir2an 708 . 2 ( I β†Ύ 𝐴):𝐴⟢On
9 fvresi 7166 . . . . 5 (π‘₯ ∈ 𝐴 β†’ (( I β†Ύ 𝐴)β€˜π‘₯) = π‘₯)
109adantr 480 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (( I β†Ύ 𝐴)β€˜π‘₯) = π‘₯)
11 fvresi 7166 . . . . 5 (𝑦 ∈ 𝐴 β†’ (( I β†Ύ 𝐴)β€˜π‘¦) = 𝑦)
1211adantl 481 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (( I β†Ύ 𝐴)β€˜π‘¦) = 𝑦)
1310, 12eleq12d 2821 . . 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 6044 . 2 dom ( I β†Ύ 𝐴) = 𝐴
168, 3, 14, 15issmo 8346 1 Smo ( I β†Ύ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943   I cid 5566  ran crn 5670   β†Ύ cres 5671  Ord word 6356  Oncon0 6357   Fn wfn 6531  βŸΆwf 6532  β€˜cfv 6536  Smo wsmo 8343
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-smo 8344
This theorem is referenced by:  smo0  8356
  Copyright terms: Public domain W3C validator