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Theorem smores3 7611
Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
smores3 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → Smo (𝐴𝐶))

Proof of Theorem smores3
StepHypRef Expression
1 dmres 5569 . . . . . 6 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 incom 3940 . . . . . 6 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
31, 2eqtri 2774 . . . . 5 dom (𝐴𝐵) = (dom 𝐴𝐵)
43eleq2i 2823 . . . 4 (𝐶 ∈ dom (𝐴𝐵) ↔ 𝐶 ∈ (dom 𝐴𝐵))
5 smores 7610 . . . 4 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ dom (𝐴𝐵)) → Smo ((𝐴𝐵) ↾ 𝐶))
64, 5sylan2br 494 . . 3 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵)) → Smo ((𝐴𝐵) ↾ 𝐶))
763adant3 1126 . 2 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → Smo ((𝐴𝐵) ↾ 𝐶))
8 inss2 3969 . . . . . 6 (dom 𝐴𝐵) ⊆ 𝐵
98sseli 3732 . . . . 5 (𝐶 ∈ (dom 𝐴𝐵) → 𝐶𝐵)
10 ordelss 5892 . . . . . 6 ((Ord 𝐵𝐶𝐵) → 𝐶𝐵)
1110ancoms 468 . . . . 5 ((𝐶𝐵 ∧ Ord 𝐵) → 𝐶𝐵)
129, 11sylan 489 . . . 4 ((𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → 𝐶𝐵)
13123adant1 1124 . . 3 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → 𝐶𝐵)
14 resabs1 5577 . . 3 (𝐶𝐵 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐶))
15 smoeq 7608 . . 3 (((𝐴𝐵) ↾ 𝐶) = (𝐴𝐶) → (Smo ((𝐴𝐵) ↾ 𝐶) ↔ Smo (𝐴𝐶)))
1613, 14, 153syl 18 . 2 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → (Smo ((𝐴𝐵) ↾ 𝐶) ↔ Smo (𝐴𝐶)))
177, 16mpbid 222 1 ((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → Smo (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1624  wcel 2131  cin 3706  wss 3707  dom cdm 5258  cres 5260  Ord word 5875  Smo wsmo 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ord 5879  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-smo 7604
This theorem is referenced by: (None)
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